TPAT-1 Tests - เชาวน์ปัญญา - [6]

Let's solve this step by step:

1. Let's say a man's work rate = M and a woman's work rate = W

2. From first scenario: (7M + 6W) × 7 days = 1 job

3. From second scenario: (5M + 4W) × 10 days = 1 job

4. Therefore: 49M + 42W = 50M + 40W 49M - 50M = 40W - 42W -M = -2W M = 2W

5. Substituting back: (7×2W + 6W) × 7 = 1 job 20W × 7 = 1 job 140W = 1 job

6. For 6 men and 2 women: (6×2W + 2W) × x days = 1 job 14W × x = 1 job x = 1/(14W) = 140W/(14W) = 10 days

{"correct_answer_key":"e"}

I need to solve this work rate problem by finding the individual work rates of men and women.

Let's denote:

• m = work rate of one man per day
• w = work rate of one woman per day

From the first scenario: 7m + 6w = 1/7 (of the job per day) From the second scenario: 5m + 4w = 1/10 (of the job per day)

Solving these equations: 7m + 6w = 1/7 5m + 4w = 1/10

Multiplying the first equation by 10 and the second by 7: 70m + 60w = 10/7 35m + 28w = 7/10

Subtracting the second from the first: 35m + 32w = 10/7 - 7/10

Solving for the values of m and w, and then using these to calculate how long it would take 6 men and 2 women: 6m + 2w = work rate per day 1/(6m + 2w) = number of days needed

After calculating, the answer is 12 days.

{"correct_answer_key":"c"}

I'll solve this Thai language problem about work rates.

In this problem:

• 7 men and 6 women finish a job in 7 days
• 5 men and 4 women finish the same job in 10 days
• We need to find how long it takes 6 men and 2 women to complete the job

Let me set up equations where m = work rate of one man and w = work rate of one woman:

• From the first condition: (7m + 6w) × 7 = total work
• From the second condition: (5m + 4w) × 10 = total work

Since they equal the same total work: 49m + 42w = 50m + 40w Solving: m = 2w (each man works twice as fast as a woman)

For 6 men and 2 women:

• Daily work rate = 6m + 2w = 6(2w) + 2w = 14w
• Total work = 140w (from either original condition)
• Days needed = 140w ÷ 14w = 10 days

{"correct_answer_key":"e"}

Let me solve this work problem step by step.

Let's denote:

• Work rate of 1 man = m (work units per day)
• Work rate of 1 woman = w (work units per day)
• Total work = W

From the given information:

1. 7 men + 6 women complete the work in 7 days: (7m + 6w) × 7 = W 49m + 42w = W

2. 5 men + 4 women complete the work in 10 days: (5m + 4w) × 10 = W 50m + 40w = W

Since both equal W: 49m + 42w = 50m + 40w 42w - 40w = 50m - 49m 2w = m

So 1 man's work rate equals 2 women's work rate.

Substituting m = 2w into the first equation: 49(2w) + 42w = W 98w + 42w = W 140w = W

Now for 6 men + 2 women: Work rate per day = 6m + 2w = 6(2w) + 2w = 12w + 2w = 14w

Time needed = W ÷ (14w) = 140w ÷ 14w = 10 days

{"correct_answer_key":"e"}

Let me solve this step by step.

This is a work rate problem. Let's denote:

• m = one man's work rate per day
• w = one woman's work rate per day
• W = total work

From the given information:

• 7 men + 6 women complete the work in 7 days: (7m + 6w) × 7 = W
• 5 men + 4 women complete the work in 10 days: (5m + 4w) × 10 = W

This gives us:

• 49m + 42w = W ... (1)
• 50m + 40w = W ... (2)

Since both equal W: 49m + 42w = 50m + 40w 2w = m

Substituting m = 2w into equation (1): 49(2w) + 42w = W 140w = W

For 6 men and 2 women: Daily work rate = 6m + 2w = 6(2w) + 2w = 14w

Days needed = W ÷ (14w) = 140w ÷ 14w = 10 days

{"correct_answer_key":"e"}

Let me solve this step by step.

Let's define:

• m = work done by 1 man per day
• w = work done by 1 woman per day
• Total work = W

From the given information:

• 7 men + 6 women complete work in 7 days: (7m + 6w) × 7 = W
• 5 men + 4 women complete work in 10 days: (5m + 4w) × 10 = W

This gives us:

• 49m + 42w = W ... (1)
• 50m + 40w = W ... (2)

From equations (1) and (2): 49m + 42w = 50m + 40w 2w = m So, 1 man's work = 2 women's work

Substituting m = 2w into equation (1): 49(2w) + 42w = W 98w + 42w = W 140w = W

Now, for 6 men + 2 women: Work per day = 6m + 2w = 6(2w) + 2w = 12w + 2w = 14w

Days needed = W ÷ 14w = 140w ÷ 14w = 10 days

{"correct_answer_key": "e"}

Let me solve this Thai math problem step by step.

First, I'll set up equations where m = work done by 1 man per day and w = work done by 1 woman per day.

From the given information:

• 7 men + 6 women complete the job in 7 days: (7m + 6w) × 7 = Total Work
• 5 men + 4 women complete the job in 10 days: (5m + 4w) × 10 = Total Work

This gives us:

• 49m + 42w = Total Work
• 50m + 40w = Total Work

Since both equal the total work: 49m + 42w = 50m + 40w 2w = m

So 1 man does the same work as 2 women.

Substituting back: Total Work = 49(2w) + 42w = 140w

For 6 men + 2 women: Daily work = 6(2w) + 2w = 14w

Days needed = 140w ÷ 14w = 10 days

{"correct_answer_key":"e"}

I need to solve this work rate problem step by step.

Let me define variables:

• Let m = work rate of one man per day
• Let w = work rate of one woman per day

From the given information:

• 7 men + 6 women complete the work in 7 days
• 5 men + 4 women complete the work in 10 days

Setting up equations:

• 7(7m + 6w) = 1 work unit → 49m + 42w = 1
• 10(5m + 4w) = 1 work unit → 50m + 40w = 1

Solving the system: From equation 1: 49m + 42w = 1 From equation 2: 50m + 40w = 1

Multiply equation 1 by 20: 980m + 840w = 20 Multiply equation 2 by 21: 1050m + 840w = 21

Subtracting: 70m = 1, so m = 1/70

Substituting back: 49(1/70) + 42w = 1 7/10 + 42w = 1 42w = 3/10 w = 3/420 = 1/140

Now for 6 men + 2 women: Daily work rate = 6m + 2w = 6(1/70) + 2(1/140) = 6/70 + 2/140 = 6/70 + 1/70 = 7/70 = 1/10

Time needed = 1 ÷ (1/10) = 10 days

{"correct_answer_key":"e"}

Looking at this Thai work rate problem, I need to find the individual work rates for men and women, then calculate how long 6 men and 2 women would take.

Let m = work rate of one man per day, w = work rate of one woman per day

From the given conditions:

• 7 men + 6 women complete job in 7 days: (7m + 6w) × 7 = 1, so 7m + 6w = 1/7
• 5 men + 4 women complete job in 10 days: (5m + 4w) × 10 = 1, so 5m + 4w = 1/10

Solving this system of equations: From manipulation of these equations, I get m = 1/70 and w = 1/140

For 6 men and 2 women: Combined work rate = 6m + 2w = 6(1/70) + 2(1/140) = 6/70 + 1/70 = 7/70 = 1/10

Since they complete 1/10 of the job per day, they need 10 days to finish.

{"correct_answer_key":"e"}

To solve this problem, we need to determine the work rates of men and women and then use those rates to find out how long it will take for 6 men and 2 women to complete the same job.

Let's denote:

• ( m ) as the work rate of one man per day.
• ( w ) as the work rate of one woman per day.

From the problem, we have two scenarios:

1. 7 men and 6 women complete the job in 7 days: [ (7m + 6w) \times 7 = 1 \text{ job} ] Simplifying, we get: [ 49m + 42w = 1 \quad \text{(Equation 1)} ]

2. 5 men and 4 women complete the job in 10 days: [ (5m + 4w) \times 10 = 1 \text{ job} ] Simplifying, we get: [ 50m + 40w = 1 \quad \text{(Equation 2)} ]

Now, we solve these two equations simultaneously. First, multiply Equation 1 by 10 and Equation 2 by 7 to align the coefficients of ( m ):

[ 490m + 420w = 10 \quad \text{(Equation 3)} ] [ 350m + 280w = 7 \quad \text{(Equation 4)} ]

Subtract Equation 4 from Equation 3:

[ (490m + 420w) - (350m + 280w) = 10 - 7 ] [ 140m + 140w = 3 ] [ 140(m + w) = 3 ] [ m + w = \frac{3}{140} ]

Now, substitute ( w = \frac{3}{140} - m ) into Equation 1:

[ 49m + 42\left(\frac{3}{140} - m\right) = 1 ] [ 49m + \frac{126}{140} - 42m = 1 ] [ 7m + \frac{126}{140} = 1 ] [ 7m + \frac{9}{10} = 1 ] [ 7m = 1 - \frac{9}{10} ] [ 7m = \frac{1}{10} ] [ m = \frac{1}{70} ]

Substitute ( m = \frac{1}{70} ) back into ( m + w = \frac{3}{140} ):

[ \frac{1}{70} + w = \frac{3}{140} ] [ w = \frac{3}{140} - \frac{2}{140} ] [ w = \frac{1}{140} ]

Now, we know:

• ( m = \frac{1}{70} )
• ( w = \frac{1}{140} )

Next, calculate the work rate of 6 men and 2 women:

[ 6m + 2w = 6 \times \frac{1}{70} + 2 \times \frac{1}{140} ] [ = \frac{6}{70} + \frac{2}{140} ] [ = \frac{12}{140} + \frac{2}{140} ] [ = \frac{14}{140} ] [ = \frac{1}{10} ]

This means 6 men and 2 women can complete (\frac{1}{10}) of the job per day. Therefore, the time required to complete the job is:

[ \text{Time} = \frac{1}{\frac{1}{10}} = 10 \text{ days} ]

Thus, the correct answer is:

{"correct_answer_key":"e"}

Explanation:

This is a work-rate problem involving men and women with different efficiencies. Here's how to solve it step-by-step:

1. Define Variables:

   • Let the work rate of 1 man per day = M (งาน/วัน)
   • Let the work rate of 1 woman per day = W (งาน/วัน)
   • Total work = 1 unit (งาน 1 ชิ้น)

2. Set Up Equations:

   • First Scenario:
     7 men + 6 women finish the work in 7 days.
     Total work done:
     ( (7M + 6W) \times 7 = 1 )
     → ( 49M + 42W = 1 ) ...(1)

   • Second Scenario:
     5 men + 4 women finish the work in 10 days.
     Total work done:
     ( (5M + 4W) \times 10 = 1 )
     → ( 50M + 40W = 1 ) ...(2)

3. Solve the Equations:

   • Subtract (1) from (2):
     ( (50M + 40W) - (49M + 42W) = 0 )
     → ( M - 2W = 0 )
     → ( M = 2W )

   • Substitute ( M = 2W ) into (1):
     ( 49(2W) + 42W = 1 )
     → ( 98W + 42W = 1 )
     → ( 140W = 1 )
     → ( W = \frac{1}{140} )

   • Then, ( M = 2W = \frac{2}{140} = \frac{1}{70} )

4. Calculate for 6 Men + 2 Women:

   • Combined work rate:
     ( 6M + 2W = 6 \left( \frac{1}{70} \right) + 2 \left( \frac{1}{140} \right) )
     ( = \frac{6}{70} + \frac{2}{140} )
     ( = \frac{12}{140} + \frac{2}{140} )
     ( = \frac{14}{140} )
     ( = \frac{1}{10} ) (งาน/วัน)

   • Time taken:
     ( \text{Time} = \frac{\text{Total Work}}{\text{Rate}} = \frac{1}{\frac{1}{10}} = 10 ) days

Answer:

The correct answer is 10 days (Option E).

{"correct_answer_key":"e"}

Explanation:
Let ( m ) be the work rate of one man per day, and ( w ) be the work rate of one woman per day. The total work is constant.

From the first scenario:
( 7m + 6w = \frac{1}{7} ) (work per day)
So total work ( = (7m + 6w) \times 7 = 49m + 42w ).

From the second scenario:
( 5m + 4w = \frac{1}{10} ) (work per day)
So total work ( = (5m + 4w) \times 10 = 50m + 40w ).

Set the total work equal:
( 49m + 42w = 50m + 40w )
( 49m + 42w - 50m - 40w = 0 )
( -m + 2w = 0 )
So ( m = 2w ).

Substitute ( m = 2w ) into the total work:
Total work ( = 49(2w) + 42w = 98w + 42w = 140w ).

Now, for 6 men and 2 women:
Work rate per day ( = 6m + 2w = 6(2w) + 2w = 12w + 2w = 14w ).

Number of days required ( = \frac{\text{Total work}}{\text{Work rate per day}} = \frac{140w}{14w} = 10 ) days.

Thus, the answer is 10 days, which corresponds to option E.

{"correct_answer_key":"e"}

Explanation:

The problem involves finding how many days it takes for 6 men and 2 women to complete a certain work, given the work rates from two scenarios.

• First scenario: 7 men and 6 women complete the work in 7 days.
• Second scenario: 5 men and 4 women complete the same work in 10 days.

To solve, define the work rates:

• Let (M) be the work rate of one man per day (fraction of work completed per day).
• Let (W) be the work rate of one woman per day.

From the first scenario:
[ (7M + 6W) \times 7 = 1 \quad \text{(total work)}
]
Simplifying:
[ 49M + 42W = 1 \quad \text{(Equation 1)}
]

From the second scenario:
[ (5M + 4W) \times 10 = 1
]
Simplifying:
[ 50M + 40W = 1 \quad \text{(Equation 2)}
]

Solve the system of equations:

1. Divide Equation 1 by 7:
   [ 7M + 6W = \frac{1}{7} \quad \text{(A)}
   ]
2. Divide Equation 2 by 10:
   [ 5M + 4W = \frac{1}{10} \quad \text{(B)}
   ]

Use elimination:

• Multiply (A) by 2:
  [ 14M + 12W = \frac{2}{7} \quad \text{(A1)}
  ]
• Multiply (B) by 3:
  [ 15M + 12W = \frac{3}{10} \quad \text{(B1)}
  ]
• Subtract (A1) from (B1):
  [ (15M + 12W) - (14M + 12W) = \frac{3}{10} - \frac{2}{7}
  ]
  [ M = \frac{3}{10} - \frac{2}{7} = \frac{21}{70} - \frac{20}{70} = \frac{1}{70}
  ]
  So, (M = \frac{1}{70}) (one man's work rate).

Substitute (M = \frac{1}{70}) into (B):
[ 5 \left(\frac{1}{70}\right) + 4W = \frac{1}{10}
]
[ \frac{5}{70} + 4W = \frac{1}{10} \implies \frac{1}{14} + 4W = \frac{1}{10}
]
[ 4W = \frac{1}{10} - \frac{1}{14} = \frac{7}{70} - \frac{5}{70} = \frac{2}{70} = \frac{1}{35}
]
[ W = \frac{1}{35} \div 4 = \frac{1}{140}
]
So, (W = \frac{1}{140}) (one woman's work rate).

Now, for 6 men and 2 women:

• Work rate of 6 men: (6 \times \frac{1}{70} = \frac{6}{70} = \frac{3}{35})
• Work rate of 2 women: (2 \times \frac{1}{140} = \frac{2}{140} = \frac{1}{70})
• Combined work rate: (\frac{3}{35} + \frac{1}{70} = \frac{6}{70} + \frac{1}{70} = \frac{7}{70} = \frac{1}{10})

Time (T) to complete the work:
[ T = \frac{1}{\text{work rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ days}
]

Thus, it takes 10 days for 6 men and 2 women to complete the work, corresponding to option (e).

{
  "correct_answer_key": "e"
}

The problem involves finding the time for 6 men and 2 women to complete a work based on given scenarios. Let ( m ) be the work rate of one man per day and ( w ) be the work rate of one woman per day. From the first scenario (7 men and 6 women in 7 days) and the second (5 men and 4 women in 10 days), the equation ((7m + 6w) \times 7 = (5m + 4w) \times 10) leads to (m = 2w). The total work is (140w) (from first scenario). For 6 men and 2 women, the combined rate is (6m + 2w = 14w) (since (m = 2w)). Thus, time = total work / rate = (140w / 14w = 10) days. The correct option is e.

{"correct_answer_key": "e"}

This question is in Thai and asks: "7 men and 6 women can complete a job in 7 days. 5 men and 4 women can complete the same job in 10 days. How many days will it take for 6 men and 2 women to complete the same job?"

Let 'm' be the work done by one man per day and 'w' be the work done by one woman per day. We can set up a system of equations:

• 7m * 7 + 6w * 7 = 1 (49m + 42w = 1)
• 5m * 10 + 4w * 10 = 1 (50m + 40w = 1)

Solving this system of equations:

Multiply the first equation by 10: 490m + 420w = 10 Multiply the second equation by 7: 350m + 280w = 7

Subtract the second equation from the first: 140m + 140w = 3, which simplifies to m + w = 3/140

Now substitute this back into the first equation (49m + 42w = 1): 49m + 42(3/140 - m) = 1 49m + 9/10 - 42m = 1 7m = 1/10 m = 1/70

Substitute m back into m + w = 3/140: 1/70 + w = 3/140 w = 3/140 - 2/140 w = 1/140

Now, let 'x' be the number of days it takes for 6 men and 2 women to complete the job: 6(1/70)x + 2(1/140)x = 1 6x/70 + 2x/140 = 1 12x/140 + 2x/140 = 1 14x/140 = 1 x = 10

{"correct_answer_key": "e"}

Let 'm' be the amount of work a man can do in a day, and 'w' be the amount of work a woman can do in a day. Let 'W' be the total amount of work.

We can set up two equations based on the given information:

1. (7m + 6w) * 7 = W
2. (5m + 4w) * 10 = W

Since both expressions equal W, we can set them equal to each other:

(7m + 6w) * 7 = (5m + 4w) * 10 49m + 42w = 50m + 40w 2w = m

Now we know that a man does twice as much work as a woman in a day. We can substitute m = 2w into either equation to find W in terms of w. Let's use the first equation:

(7(2w) + 6w) * 7 = W (14w + 6w) * 7 = W (20w) * 7 = W 140w = W

Now we want to find how many days it takes for 6 men and 2 women to complete the same work. Let 'd' be the number of days.

(6m + 2w) * d = W Substitute m = 2w: (6(2w) + 2w) * d = 140w (12w + 2w) * d = 140w 14w * d = 140w d = 140w / 14w d = 10

Therefore, it will take 10 days for 6 men and 2 women to complete the same work.

{"correct_answer_key":"e"}

This is a work-rate problem. We need to determine the individual work rates of men and women, then calculate the time it takes for a different group to complete the same work.

Let:

• 'm' be the work rate of a man per day
• 'w' be the work rate of a woman per day
• 'W' be the total amount of work

From the problem statement, we have two equations:

1. 7m + 6w = W/7 (7 men and 6 women complete the work in 7 days)
2. 5m + 4w = W/10 (5 men and 4 women complete the work in 10 days)

We can solve these equations to find the values of 'm' and 'w' in terms of 'W'. Multiply equation 1 by 10 and equation 2 by 7:

1. 70m + 60w = 10W/7
2. 35m + 28w = 7W/10

Multiply equation 1 by 7 and equation 2 by 10:

1. 490m + 420w = 10W
2. 350m + 280w = 7W

Multiply equation 1 by 5 and equation 2 by 7:

1. 35m + 30w = 5W/7
2. 35m + 28w = 7W/10

Subtract the second equation from the first:

2w = 5W/7 - 7W/10 = (50W - 49W)/70 = W/70 w = W/140

Substitute w = W/140 into the first equation:

7m + 6(W/140) = W/7 7m + 3W/70 = W/7 7m = W/7 - 3W/70 = (10W - 3W)/70 = 7W/70 m = W/70

Now we know the work rate of a man is W/70 and the work rate of a woman is W/140.

We want to find the number of days (d) it takes for 6 men and 2 women to complete the work:

6m + 2w = W/d 6(W/70) + 2(W/140) = W/d 6W/70 + 2W/140 = W/d 12W/140 + 2W/140 = W/d 14W/140 = W/d W/10 = W/d d = 10

Therefore, it will take 10 days for 6 men and 2 women to complete the work.

{"correct_answer_key":"e"}

Let M be the work rate of one man and W be the work rate of one woman. The total work (T) is constant.

From the first condition: 7 men and 6 women complete the work in 7 days. So, T = (7M + 6W) * 7 T = 49M + 42W (Equation 1)

From the second condition: 5 men and 4 women complete the work in 10 days. So, T = (5M + 4W) * 10 T = 50M + 40W (Equation 2)

Equating Equation 1 and Equation 2: 49M + 42W = 50M + 40W 42W - 40W = 50M - 49M 2W = M

This means one man's work rate is equivalent to two women's work rate.

Now, substitute M = 2W into Equation 1 to find the total work in terms of W: T = 49(2W) + 42W T = 98W + 42W T = 140W

Finally, we want to find how many days it takes for 6 men and 2 women to complete the work. Let D be the number of days. The combined work rate of 6 men and 2 women is: 6M + 2W = 6(2W) + 2W = 12W + 2W = 14W

Now, use the formula T = (Work Rate) * D: 140W = (14W) * D D = 140W / 14W D = 10

Therefore, 6 men and 2 women will complete the work in 10 days.

{"correct_answer_key":"e"}

Let M be the work rate of one man and W be the work rate of one woman. The total work done is constant.

From the first statement: 7 men and 6 women complete the work in 7 days. Work = (7M + 6W) * 7 (Equation 1)

From the second statement: 5 men and 4 women complete the work in 10 days. Work = (5M + 4W) * 10 (Equation 2)

Since the work is the same, we can equate Equation 1 and Equation 2: (7M + 6W) * 7 = (5M + 4W) * 10 49M + 42W = 50M + 40W 42W - 40W = 50M - 49M 2W = M

This means 1 man does the work of 2 women. We can substitute M = 2W into one of the work equations. Let's use Equation 2: Work = (5(2W) + 4W) * 10 Work = (10W + 4W) * 10 Work = 14W * 10 Work = 140W

Now, we want to find out how many days it takes for 6 men and 2 women to complete the same work. Let D be the number of days. Work = (6M + 2W) * D

Substitute M = 2W into this equation: Work = (6(2W) + 2W) * D Work = (12W + 2W) * D Work = 14W * D

Now, equate this with the total work we found (140W): 14W * D = 140W Divide both sides by 14W: D = 140W / 14W D = 10

So, 6 men and 2 women will complete the work in 10 days.

{"correct_answer_key":"e"}

กำหนดให้ m คืออัตราการทำงานของผู้ชาย 1 คนต่อวัน และ f คืออัตราการทำงานของผู้หญิง 1 คนต่อวัน จากโจทย์จะได้สมการ (7m + 6f) * 7 = (5m + 4f) * 10 เมื่อแก้สมการจะได้ว่า m = 2f หมายความว่าผู้ชาย 1 คนทำงานได้เท่ากับผู้หญิง 2 คน ปริมาณงานทั้งหมดคือ (5(2f) + 4f) * 10 = 140f ดังนั้น ถ้าใช้ชาย 6 คน และหญิง 2 คน ซึ่งมีกำลังงานเท่ากับ 6(2f) + 2f = 14f จะใช้เวลาทำงานทั้งหมด 140f / 14f = 10 วัน

{"correct_answer_key":"e"}

This is a work-rate problem. Let 'm' be the work rate of a man and 'w' be the work rate of a woman. The total work done is constant. We can set up two equations based on the given information:

• 7m + 6w = 1/7 (Work done in 7 days)
• 5m + 4w = 1/10 (Work done in 10 days)

We need to solve for 'm' and 'w'. Multiplying the first equation by 2 and the second by 3, we get:

• 14m + 12w = 2/7
• 15m + 12w = 3/10

Subtracting the first equation from the second:

m = 3/10 - 2/7 = (21 - 20) / 70 = 1/70

Substituting m = 1/70 into the first equation:

7(1/70) + 6w = 1/7 1/10 + 6w = 1/7 6w = 1/7 - 1/10 = (10 - 7) / 70 = 3/70 w = 3/70 * (1/6) = 1/140

Now we need to find how many days (let's call it 'd') it takes for 6 men and 2 women to complete the work:

6m + 2w = 1/d 6(1/70) + 2(1/140) = 1/d 6/70 + 2/140 = 1/d 12/140 + 2/140 = 1/d 14/140 = 1/d 1/10 = 1/d d = 10

Therefore, 6 men and 2 women will complete the work in 10 days.

{"correct_answer_key":"e"}

The problem involves finding the time taken for a group of men and women to complete a job based on given scenarios. From the first scenario, 7 men and 6 women finish the job in 7 days, and from the second, 5 men and 4 women finish it in 10 days. By setting up equations for the work rates, we find that one man's work rate is ( \frac{1}{70} ) of the job per day, and one woman's work rate is ( \frac{1}{140} ) of the job per day. For 6 men and 2 women, the combined work rate is ( 6 \times \frac{1}{70} + 2 \times \frac{1}{140} = \frac{6}{70} + \frac{2}{140} = \frac{12}{140} + \frac{2}{140} = \frac{14}{140} = \frac{1}{10} ) of the job per day. Thus, the time to complete the job is ( \frac{1}{\frac{1}{10}} = 10 ) days, corresponding to option e.

{"correct_answer_key":"e"}

To solve this problem, we need to determine the time required for 6 men and 2 women to complete the same work, given the productivity rates from two scenarios. Let ( M ) represent the work done by one man per day, and ( W ) represent the work done by one woman per day.

Step 1: Set up equations from the given scenarios

• First scenario: 7 men and 6 women complete the work in 7 days.
  Total work = ( (7M + 6W) \times 7 ).
• Second scenario: 5 men and 4 women complete the work in 10 days.
  Total work = ( (5M + 4W) \times 10 ).

Since the work is the same in both cases:
[ (7M + 6W) \times 7 = (5M + 4W) \times 10 ]

Step 2: Solve for the relationship between ( M ) and ( W )

Expand the equation:
[ 49M + 42W = 50M + 40W ]
Rearrange terms:
[ 49M - 50M + 42W - 40W = 0 ]
[ -M + 2W = 0 ]
[ M = 2W ]
This means one man is twice as productive as one woman.

Step 3: Calculate the total work

Using the first scenario and substituting ( M = 2W ):
[ 7M + 6W = 7(2W) + 6W = 14W + 6W = 20W ]
Total work = ( 20W \times 7 = 140W ) (woman-days).
Verification with the second scenario:
[ 5M + 4W = 5(2W) + 4W = 10W + 4W = 14W ]
Total work = ( 14W \times 10 = 140W ) (consistent).

Step 4: Find the time for 6 men and 2 women

Convert the team to woman-equivalents using ( M = 2W ):

• 6 men = ( 6 \times 2W = 12W )
• 2 women = ( 2W )
• Total productivity per day = ( 12W + 2W = 14W )

Let ( D ) be the days required:
[ 14W \times D = 140W ]
[ D = \frac{140W}{14W} = 10 ]

Conclusion

6 men and 2 women will complete the work in 10 days. The correct answer key is e.

{"correct_answer_key":"e"}

อธิบาย:
กำหนดให้กำลังการทำงานของชาย 1 คนต่อวัน = x, หญิง 1 คนต่อวัน = y
งานทั้งหมด = W

จากโจทย์

1. 7x + 6y = W/7
2. 5x + 4y = W/10

คูณสมการแรกด้วย 7:
7(7x + 6y) = W
49x + 42y = W

คูณสมการที่สองด้วย 10:
10(5x + 4y) = W
50x + 40y = W

ตั้งสมการ
49x + 42y = 50x + 40y
ย้ายข้าง
50x - 49x + 40y - 42y = 0
x - 2y = 0
x = 2y

แทนค่า x = 2y ในสมการแรก:
7x + 6y = W/7
7(2y) + 6y = W/7
14y + 6y = W/7
20y = W/7
W = 140y

ถ้าใช้ชาย 6 คน หญิง 2 คน
ต่อวันทำได้ = 6x + 2y = 6(2y) + 2y = 12y + 2y = 14y
จำนวนวันที่ใช้ = งานทั้งหมด / งานที่ทำได้ต่อวัน = 140y / 14y = 10

ดังนั้น คำตอบคือข้อ e

{"correct_answer_key":"e"}

สมมติให้ประสิทธิภาพการทำงานของชาย 1 คนต่อวัน = x และหญิง 1 คนต่อวัน = y

จากโจทย์: ชาย 7 คน + หญิง 6 คน ทำงานเสร็จใน 7 วัน => (7x + 6y) * 7 = 1 งาน => 7x + 6y = 1/7

ชาย 5 คน + หญิง 4 คน ทำงานเสร็จใน 10 วัน => (5x + 4y) * 10 = 1 งาน => 5x + 4y = 1/10

แก้สมการสองตัวนี้เพื่อหา x และ y: จากสมการแรก: 7x + 6y = 1/7 จากสมการที่สอง: 5x + 4y = 1/10

คูณสมการที่สองด้วย 3: 15x + 12y = 3/10

คูณสมการแรกด้วย 2: 14x + 12y = 2/7

ลบสมการที่สองจากสมการแรก: (15x - 14x) + (12y - 12y) = 3/10 - 2/7 => x = 3/10 - 2/7 = (21/70) - (20/70) = 1/70

แทนค่า x = 1/70 ลงในสมการ 7x + 6y = 1/7: 7*(1/70) + 6y = 1/7 => 1/10 + 6y = 1/7 => 6y = 1/7 - 1/10 = (10/70) - (7/70) = 3/70 => y = 3/70 ÷ 6 = 3/420 = 1/140

ดังนั้น: x = 1/70 (งานต่อวันของชาย 1 คน) y = 1/140 (งานต่อวันของหญิง 1 คน)

ชาย 6 คน และ หญิง 2 คน ทำงานต่อวันได้: 6x + 2y = 6*(1/70) + 2*(1/140) = 6/70 + 2/140 = 6/70 + 1/70 = 7/70 = 1/10

ดังนั้น จำนวนวันที่ใช้ทำงาน = 1 / (งานต่อวัน) = 1 / (1/10) = 10 วัน

คำตอบคือ 10 วัน ซึ่งตรงกับตัวเลือก e.

{"correct_answer_key":"e"}

จากข้อมูล ชาย 7 คนและหญิง 6 คนทำงานเสร็จใน 7 วัน หมายความว่าความสามารถรวมของชายและหญิงต่อวันคือ (7+6) งาน/วัน × 7 วัน = 49 งาน (สมมติ 1 งานเป็นงานเดียวกัน) ต่อมา ชาย 5 คนและหญิง 4 คนทำงานเสร็จใน 10 วัน ความสามารถรวมของพวกเขาคือ (5+4) งาน/วัน × 10 วัน = 45 งาน ซึ่งใกล้เคียงกันแสดงว่าความสามารถของชายและหญิงต่อวันเป็นค่าคงที่

สมมติให้ความสามารถของชายต่อวันเป็น x และของหญิงต่อวันเป็น y

จากข้อมูล: 7x + 6y = งานต่อวัน (รวม) = 1/7 งาน/วัน 5x + 4y = 1/10 งาน/วัน

แก้สมการ: 7x + 6y = 1/7 5x + 4y = 1/10

คูณสมการแรกด้วย 10 และสมการที่สองด้วย 7 เพื่อกำจัดตัวแปร: (7x + 6y) × 10 = 10/7 (5x + 4y) × 7 = 7/10

แต่เพื่อความง่ายขึ้น เราสามารถแก้สมการโดยใช้วิธีการแทนที่หรือการคูณข้าม

จากสมการแรก: 7x + 6y = 1/7

จากสมการที่สอง: 5x + 4y = 1/10

แก้สมการนี้เพื่อหา x และ y แล้วคำนวณเวลาที่ชาย 6 คนและหญิง 2 คนจะทำงานให้เสร็จ

แทนค่า: จากสมการแรก: 7x + 6y = 1/7

จากสมการที่สอง: 5x + 4y = 1/10

แก้สมการโดยใช้วิธีการแทนที่:

คูณสมการแรกด้วย 4: 28x + 24y = 4/7

คูณสมการที่สองด้วย 6: 30x + 24y = 6/10 = 3/5

ลบสมการ: (28x + 24y) - (30x + 24y) = 4/7 - 3/5 -2x = (4/7) - (3/5)

คำนวณ: (4/7) - (3/5) = (20/35) - (21/35) = -1/35

ดังนั้น: -2x = -1/35 x = 1/70

แทนค่า x กลับเข้าไปในสมการแรก: 7(1/70) + 6y = 1/7 1/10 + 6y = 1/7 6y = 1/7 - 1/10 6y = (10/70) - (7/70) = 3/70 y = (3/70) / 6 = 3/70 × 1/6 = 1/140

ตอนนี้เรามีความสามารถต่อวันของชายและหญิง: ชาย: x = 1/70 งาน/วัน หญิง: y = 1/140 งาน/วัน

ต้องการให้ชาย 6 คนและหญิง 2 คนทำงานให้เสร็จ: ความสามารถรวม = 6 × 1/70 + 2 × 1/140 = 6/70 + 2/140

คำนวณ: 6/70 = 3/35 2/140 = 1/70

รวม: 3/35 + 1/70 = (6/70) + (1/70) = 7/70 = 1/10 งาน/วัน

ดังนั้น เวลาที่จะทำงานเสร็จ = 1 งาน / (1/10 งาน/วัน) = 10 วัน

คำตอบคือ 10 วัน

{"correct_answer_key":"e"}

ขั้นแรก เราจะสมมติให้ชาย 1 คนทำงานได้วันละ ( x ) หน่วย และหญิง 1 คนทำงานได้วันละ ( y ) หน่วย

จากโจทย์:

• ชาย 7 คน หญิง 6 คน ทำงานเสร็จใน 7 วัน: [ 7(7x + 6y) = 49x + 42y ]

• ชาย 5 คน หญิง 4 คน ทำงานเสร็จใน 10 วัน: [ 10(5x + 4y) = 50x + 40y ]

เนื่องจากเป็นงานเดียวกัน ดังนั้น: [ 49x + 42y = 50x + 40y ]

จัดรูปสมการใหม่: [ 50x - 49x = 42y - 40y \ x = 2y ]

ดังนั้น อัตราการทำงานของชาย 1 คน เท่ากับหญิง 2 คน (( x = 2y ))

แทนค่า ( x = 2y ) ลงในสมการแรก (หรือสมการที่สองก็ได้) เพื่อหาปริมาณงานทั้งหมด (หน่วยงาน):

• ใช้สมการแรก: [ 49x + 42y = 49(2y) + 42y = 98y + 42y = 140y ]

ดังนั้น งานทั้งหมดคือ (140y) หน่วย

โจทย์ถามว่า ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จในกี่วัน:

• ชาย 6 คน = (6x = 6(2y) = 12y)
• หญิง 2 คน = (2y)

รวมกันทำงานได้วันละ (12y + 2y = 14y) หน่วยต่อวัน

ดังนั้น จำนวนวันที่ใช้คือ: [ \frac{140y}{14y} = 10 \text{ วัน} ]

คำตอบที่ถูกต้องคือข้อ e (10 วัน)

{"correct_answer_key":"e"}

ในการแก้ปัญหานี้ เราต้องหาความสามารถในการทำงานของชายและหญิงก่อน:

1. ชาย 7 คน และหญิง 6 คน ทำงานเสร็จใน 7 วัน
2. ชาย 5 คน และหญิง 4 คน ทำงานเสร็จใน 10 วัน

ให้ ( M ) แทนความสามารถในการทำงานของชาย 1 คนต่อวัน และ ( W ) แทนความสามารถในการทำงานของหญิง 1 คนต่อวัน

จากข้อมูล:

• ( 7M + 6W = \frac{1}{7} ) (งานทั้งหมดเสร็จใน 7 วัน)
• ( 5M + 4W = \frac{1}{10} ) (งานทั้งหมดเสร็จใน 10 วัน)

เราต้องการหาว่า ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จในกี่วัน:

• ( 6M + 2W = \frac{1}{x} )

แก้สมการเพื่อหา ( x ):

จากสมการที่ 1 และ 2:

• ( 7M + 6W = \frac{1}{7} )
• ( 5M + 4W = \frac{1}{10} )

แก้ระบบสมการนี้เพื่อหา ( M ) และ ( W ) แล้วแทนค่าในสมการ ( 6M + 2W = \frac{1}{x} ) เพื่อหา ( x ).

หลังจากคำนวณจะพบว่า ( x = 14 ).

ดังนั้น ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จใน 14 วัน ซึ่งตรงกับตัวเลือก A.

{"correct_answer_key":"a"}

ในการแก้ปัญหานี้ เราต้องคำนวณอัตราการทำงานของชายและหญิงจากข้อมูลที่ให้มา:

1. ชาย 7 คน และหญิง 6 คน ทำงานเสร็จใน 7 วัน

   • งานทั้งหมด = (7 คน + 6 คน) * 7 วัน = 91 คน-วัน

2. ชาย 5 คน และหญิง 4 คน ทำงานเสร็จใน 10 วัน

   • งานทั้งหมด = (5 คน + 4 คน) * 10 วัน = 90 คน-วัน

จากข้อมูลนี้ เราสามารถหาค่าอัตราการทำงานของชายและหญิงได้

ให้ชาย 6 คน และหญิง 2 คน ทำงาน:

• อัตราการทำงานรวม = (6 คน + 2 คน) = 8 คน

เราสามารถใช้การคำนวณเพื่อหาว่าพวกเขาจะใช้เวลากี่วันในการทำงานให้เสร็จ

จากการคำนวณจะพบว่า ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จภายใน 12 วัน

ดังนั้นคำตอบที่ถูกต้องคือ 12 วัน ซึ่งตรงกับตัวเลือก C

{"correct_answer_key":"c"}

ให้ m = อัตราทำงานของผู้ชาย/วัน, w = ของผู้หญิง. จากโจทย์: 7m+6w=1/7 และ 5m+4w=1/10 แก้ได้ m=1/70, w=1/140 ดังนั้น 6m+2w = 1/10 ใช้เวลา 10 วัน จึงตอบ e

{"correct_answer_key":"e"}

สมมติ อัตราทำงานต่อวันของชาย = m และหญิง = f
จากข้อมูล: (7m+6f)*7 = 1 → 49m+42f=1 และ (5m+4f)*10 =1 → 50m+40f=1
จากสองสมการได้ m = 2f, แทนกลับได้ f = 1/140, ดังนั้น m = 1/70
อัตราของ 6 ชาย + 2 หญิง = 6m+2f = 6/70 + 2/140 = 1/10 ดังนั้นใช้เวลา 10 วัน → ข้อ e

{"correct_answer_key":"e"}

คำอธิบายสั้นๆ: กำหนดอัตราการผลิตของคนงานชายเป็น r_m และหญิงเป็น r_w ตามสมการ (7r_m+6r_w)×7 = W และ (5r_m+4r_w)×10 = W ทำให้ r_m = 2r_w จากนั้นรวมคนงาน 6 ผู้ชายและ 2 ผู้หญิงเป็น 6r_m+2r_w = 14r_w จะใช้เวลา D ในการทำงาน W = 140r_w ดังนั้น D = W/(14r_w) = 10 วัน

{"correct_answer_key":"e"}

The total work (W) can be expressed using the combined daily rates of men ((m)) and women ((w)).

1. From the first group: ((7m+6w)\times7 = W)
2. From the second group: ((5m+4w)\times10 = W)

Setting them equal: [ 7(7m+6w)=10(5m+4w) \ 49m+42w = 50m+40w \ m = 2w ]

So each man works twice as fast as a woman.

Compute the total work using the first scenario: [ 7m+6w = 7(2w)+6w = 20w \ W = 20w \times 7 = 140w ]

With 6 men and 2 women the combined rate is: [ 6m+2w = 6(2w)+2w = 14w ]

Days required: [ \text{Days} = \frac{W}{\text{rate}} = \frac{140w}{14w} = 10 ]

Thus the correct answer is 10 days, which corresponds to option e.

{"correct_answer_key":"e"}

The combined work rate of 6 men and 2 women is calculated as follows:

• Rate of one man = ( \frac{1}{70} ) job per day
• Rate of one woman = ( \frac{1}{140} ) job per day

Combined rate:
( 6 \times \frac{1}{70} + 2 \times \frac{1}{140} = \frac{6}{70} + \frac{2}{140} = \frac{3}{35} + \frac{1}{70} = \frac{7}{70} = \frac{1}{10} ) job per day.

Thus, they finish the job in ( \frac{1}{\frac{1}{10}} = 10 ) days.

{"correct_answer_key":"e"}

Let's solve this problem step by step. We need to determine how many days it will take for 6 men and 2 women to complete the same work, given the data about different groups of men and women completing the work in different time periods.

First, let's calculate the work rates of men and women. Let's assume the work done by one man per day is (M) and by one woman per day is (W). The total work is constant, so we can set up equations based on the given scenarios.

1. 7 men and 6 women complete the work in 7 days: [ (7M + 6W) \times 7 = \text{Total Work} ] So, Total Work = (49M + 42W).

2. 5 men and 4 women complete the work in 10 days: [ (5M + 4W) \times 10 = \text{Total Work} ] So, Total Work = (50M + 40W).

Since both expressions equal the total work, we can set them equal to each other: [ 49M + 42W = 50M + 40W ] Simplifying: [ 49M - 50M + 42W - 40W = 0 ] [ -M + 2W = 0 ] [ 2W = M ] So, the work rate of one man ((M)) is twice the work rate of one woman ((W)). That is, (M = 2W).

Now, substitute (M = 2W) into one of the total work equations to find the total work. Using the first equation: [ \text{Total Work} = 49M + 42W = 49(2W) + 42W = 98W + 42W = 140W ] So, the total work is equivalent to 140 units of work done by one woman.

Next, we need to find out how many days it takes for 6 men and 2 women to complete this work. First, calculate their combined work rate per day:

• Work rate of 6 men = (6M = 6 \times 2W = 12W)
• Work rate of 2 women = (2W)
• Total work rate per day = (12W + 2W = 14W)

Now, the number of days ((D)) required to complete the total work of (140W) is: [ D = \frac{\text{Total Work}}{\text{Work rate per day}} = \frac{140W}{14W} = 10 \text{ days} ]

Thus, 6 men and 2 women will complete the work in 10 days, which corresponds to option E.

{"correct_answer_key":"e"}

Brief Explanation:

This is a work rate problem. We have two scenarios:

• 7 men and 6 women complete the job in 7 days.
• 5 men and 4 women complete the same job in 10 days.

Let M be the work rate of one man (job per day) and W be the work rate of one woman. From the equations:

• (7M + 6W) × 7 = 1 job → 49M + 42W = 1
• (5M + 4W) × 10 = 1 job → 50M + 40W = 1

Solving these, we get M = 1/70 and W = 1/140.

For 6 men and 2 women, their combined work rate is 6M + 2W = 6(1/70) + 2(1/140) = 1/10 job per day.

Thus, time to complete 1 job = 1 / (1/10) = 10 days.

The correct answer is option E.

{"correct_answer_key":"e"}

Brief Explanation:

This is a work rate problem. From the given information, we determine that 7 men and 6 women complete the job in 7 days, so their combined work rate is 1/7 jobs per day. Similarly, 5 men and 4 women complete it in 10 days, so their work rate is 1/10 jobs per day. Solving the equations, the work rate of one man is 1/70 jobs per day, and one woman is 1/140 jobs per day. For 6 men and 2 women, their combined work rate is 6*(1/70) + 2*(1/140) = 1/10 jobs per day, meaning they take 10 days to complete the job.

{"correct_answer_key":"e"}

Let's solve this problem step by step. We need to determine how many days it will take for 6 men and 2 women to complete the same job, given the work rates derived from the provided scenarios.

First, let's calculate the work rates of men and women. Let the work rate of one man be (M) (work per day) and that of one woman be (W) (work per day). The total work for the job is constant, so we can set up equations based on the given data.

• Scenario 1: 7 men and 6 women complete the job in 7 days. So, total work = ((7M + 6W) \times 7).
• Scenario 2: 5 men and 4 women complete the job in 10 days. So, total work = ((5M + 4W) \times 10).

Since the total work is the same, we equate the two expressions: ((7M + 6W) \times 7 = (5M + 4W) \times 10)

Simplify the equation: (49M + 42W = 50M + 40W) (49M - 50M + 42W - 40W = 0) (-M + 2W = 0) (2W = M)

Thus, the work rate of one man ((M)) is twice that of one woman ((W)). So, (M = 2W).

Now, let's find the total work in terms of (W). Using Scenario 1: Total work = ((7M + 6W) \times 7 = (7 \times 2W + 6W) \times 7 = (14W + 6W) \times 7 = 20W \times 7 = 140W).

Next, we calculate the combined work rate of 6 men and 2 women: Work rate of 6 men = (6M = 6 \times 2W = 12W) Work rate of 2 women = (2W) Total work rate = (12W + 2W = 14W) per day.

Finally, the number of days required to complete the total work of (140W) at a rate of (14W) per day is: Days = Total work / Work rate per day = (140W / 14W = 10) days.

Therefore, 6 men and 2 women will complete the job in 10 days, which corresponds to option E.

{"correct_answer_key":"e"}

Explanation: Let M be a man's daily work rate and W a woman's. From the given, 49M + 42W = 1 and 50M + 40W = 1. Solving yields M = 1/70 and W = 1/140. For 6 men and 2 women, rate is 6/70 + 2/140 = 1/10 job/day, so time is 10 days.

{"correct_answer_key":"e"}

Let the work rate of one man be M (jobs per day) and one woman be W.

From the first condition:
7M + 6W = 1/7 (since 7 men and 6 women finish the job in 7 days).

From the second condition:
5M + 4W = 1/10.

Solve the system:

1. 7M + 6W = 1/7
2. 5M + 4W = 1/10

Multiply (1) by 2 and (2) by 3 to eliminate W:

14M + 12W = 2/7
15M + 12W = 3/10

Subtract the first from the second:

M = 3/10 − 2/7 = (21 − 20)/70 = 1/70.

Substitute M = 1/70 into (2):

5(1/70) + 4W = 1/10
1/14 + 4W = 1/10
4W = 1/10 − 1/14 = (7 − 5)/70 = 2/70 = 1/35
W = 1/140.

Now compute the combined rate for 6 men and 2 women:

6M + 2W = 6(1/70) + 2(1/140) = 6/70 + 2/140 = 6/70 + 1/70 = 7/70 = 1/10.

Therefore, they will finish the job in 10 days.

{"correct_answer_key":"e"}

เริ่มต้นด้วยการวิเคราะห์ข้อมูลที่ให้มา:

1. ชาย 7 คน หญิง 6 คน ทำงานเสร็จใน 7 วัน
2. ชาย 5 คน หญิง 4 คน ทำงานเสร็จใน 10 วัน

สมมติว่างานที่ทำได้โดยชาย 1 คนใน 1 วัน คือ M และงานที่ทำได้โดยหญิง 1 คนใน 1 วัน คือ F

จากข้อมูลข้างต้น เราสามารถเขียนสมการได้ดังนี้:

7M + 6F = 1/7 (งานทั้งหมดใน 1 วัน) 5M + 4F = 1/10 (งานทั้งหมดใน 1 วัน)

เนื่องจากงานทั้งหมดเท่ากัน เราจึงสามารถตั้งสมการได้:

7M + 6F = 5M + 4F * (10/7)

ลดความซับซ้อนของสมการ:

7M + 6F = (50M + 40F) / 7 49M + 42F = 50M + 40F 42F - 40F = 50M - 49M 2F = M

จากนั้น เราสามารถแทนค่า M ในสมการแรก:

7M + 6F = 1/7 7(2F) + 6F = 1/7 14F + 6F = 1/7 20F = 1/7 F = 1/(7*20) F = 1/140

และ M = 2F = 2/(7*20) = 1/70

ทีนี้ มาดูกรณีที่ชาย 6 คน และหญิง 2 คน:

6M + 2F = 6*(1/70) + 2*(1/140) = 6/70 + 1/70 = 7/70 = 1/10

ดังนั้น ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จใน 10 วัน

{"correct_answer_key":"e"}

To solve this, let's denote the work rate of one man as $m$ and the work rate of one woman as $w$. The total work is denoted as $T$.

From the given information, we can form the following equations based on the work done:

1. $(7m + 6w) \times 7 = T$
2. $(5m + 4w) \times 10 = T$

Since both expressions equal $T$, we can set them equal to each other: $(7m + 6w) \times 7 = (5m + 4w) \times 10$

Expanding both sides gives: $49m + 42w = 50m + 40w$

Rearranging the equation to solve for $m$ in terms of $w$: $42w - 40w = 50m - 49m$ $2w = m$

Now, let's express $T$ using the first equation and substitute $m = 2w$: $T = (7(2w) + 6w) \times 7$ $T = (14w + 6w) \times 7$ $T = 20w \times 7$ $T = 140w$

For 6 men and 2 women, the combined work rate is $6m + 2w$. Substituting $m = 2w$: $6(2w) + 2w = 12w + 2w = 14w$

The time taken for 6 men and 2 women to complete $T$ is given by $T / (14w)$: $Time = \frac{T}{14w} = \frac{140w}{14w} = 10$ days

Therefore, it will take 10 days for 6 men and 2 women to complete the work.

{"correct_answer_key":"e"}

Let's analyze the situation:

1. 7 men and 6 women complete the job in 7 days.
2. 5 men and 4 women complete the job in 10 days.

Let's assume the work rate of 1 man is M and 1 woman is W.

From (1), the total work done is (7M + 6W) * 7.

From (2), the total work done is (5M + 4W) * 10.

Since the total work done is the same, we can equate:

(7M + 6W) * 7 = (5M + 4W) * 10

Expanding and simplifying:

49M + 42W = 50M + 40W

Rearranging:

M = 2W

Now, let's find the work rate of 6 men and 2 women:

6M + 2W = 6(2W) + 2W = 14W

The total work done is (7M + 6W) * 7 = (14W + 6W) * 7 = 140W

The number of days required for 6 men and 2 women to complete the job:

140W / 14W = 10

So, ชาย6 คน และหญิง2 คน จะทำงานแห่งเดียวกันเสร็จภายใน10 วัน.

{"correct_answer_key":"e"}

To solve this problem, we need to determine the work rates of men and women based on the given information and then calculate the time it would take for a different combination of workers to complete the same task.

Let's denote:

• ( M ) as the work rate of one man per day
• ( W ) as the work rate of one woman per day

From the first scenario:

• 7 men and 6 women complete the task in 7 days. This can be written as: [ 7M + 6W = \frac{1}{7} ]

From the second scenario:

• 5 men and 4 women complete the task in 10 days. This can be written as: [ 5M + 4W = \frac{1}{10} ]

We now have two equations:

1. ( 7M + 6W = \frac{1}{7} )
2. ( 5M + 4W = \frac{1}{10} )

To solve these equations, we can multiply the first equation by 7 and the second equation by 10 to eliminate the fractions:

1. ( 49M + 42W = 1 )
2. ( 50M + 40W = 1 )

Subtract the first equation from the second equation: [ 50M + 40W - (49M + 42W) = 1 - 1 ] [ 50M + 40W - 49M - 42W = 0 ] [ M - 2W = 0 ] [ M = 2W ]

Substitute ( M = 2W ) back into the first equation: [ 49(2W) + 42W = 1 ] [ 98W + 42W = 1 ] [ 140W = 1 ] [ W = \frac{1}{140} ]

Now, substituting ( W = \frac{1}{140} ) into ( M = 2W ): [ M = 2 \times \frac{1}{140} ] [ M = \frac{1}{70} ]

Now we need to find out how long it would take for 6 men and 2 women to complete the task: [ 6M + 2W ] [ 6 \times \frac{1}{70} + 2 \times \frac{1}{140} ] [ \frac{6}{70} + \frac{2}{140} ] [ \frac{6}{70} + \frac{1}{70} ] [ \frac{7}{70} ]

The work rate of 6 men and 2 women is ( \frac{7}{70} ) of the task per day. To find the number of days ( D ) it takes to complete the task: [ \frac{7}{70} \times D = 1 ] [ D = \frac{70}{7} ] [ D = 10 ]

Therefore, 6 men and 2 women will complete the task in 10 days, which corresponds to option E.

{"correct_answer_key":"e"}

To solve this problem, we need to determine the work rates of men and women and then use those rates to find out how long it will take for 6 men and 2 women to complete the job.

Let's denote the work rate of one man as ( m ) and the work rate of one woman as ( w ). The total work done is the same in both scenarios, so we can set up the following equations based on the given information:

1. ( 7m + 7w = 1 ) (7 men and 7 women complete the job in 7 days)
2. ( 5m + 4w = \frac{1}{10} ) (5 men and 4 women complete the job in 10 days)

From the first equation, we can simplify to: [ m + w = \frac{1}{7} ]

From the second equation, we can multiply through by 10 to get: [ 50m + 40w = 1 ]

Now we have a system of linear equations: [ m + w = \frac{1}{7} ] [ 50m + 40w = 1 ]

We can solve this system by substitution or elimination. Let's use substitution. From the first equation, we can express ( w ) in terms of ( m ): [ w = \frac{1}{7} - m ]

Substitute this into the second equation: [ 50m + 40\left(\frac{1}{7} - m\right) = 1 ] [ 50m + \frac{40}{7} - 40m = 1 ] [ 10m + \frac{40}{7} = 1 ] [ 10m = 1 - \frac{40}{7} ] [ 10m = \frac{7}{7} - \frac{40}{7} ] [ 10m = \frac{-33}{7} ] [ m = \frac{-33}{70} ]

This seems incorrect, so let's re-evaluate the substitution step. Instead, let's solve the system using elimination. Multiply the first equation by 40: [ 40m + 40w = \frac{40}{7} ]

Now subtract the second equation from this result: [ 40m + 40w - (50m + 40w) = \frac{40}{7} - 1 ] [ -10m = \frac{40}{7} - \frac{7}{7} ] [ -10m = \frac{33}{7} ] [ m = \frac{-33}{70} ]

This still seems incorrect, so let's re-evaluate the problem setup. The correct approach is to solve the system correctly:

From ( m + w = \frac{1}{7} ), we get ( w = \frac{1}{7} - m ). Substitute into the second equation: [ 50m + 40\left(\frac{1}{7} - m\right) = 1 ] [ 50m + \frac{40}{7} - 40m = 1 ] [ 10m + \frac{40}{7} = 1 ] [ 10m = 1 - \frac{40}{7} ] [ 10m = \frac{7}{7} - \frac{40}{7} ] [ 10m = \frac{-33}{7} ] [ m = \frac{-33}{70} ]

This is still incorrect, so let's use the correct approach:

From ( m + w = \frac{1}{7} ), we get ( w = \frac{1}{7} - m ). Substitute into the second equation: [ 50m + 40\left(\frac{1}{7} - m\right) = 1 ] [ 50m + \frac{40}{7} - 40m = 1 ] [ 10m + \frac{40}{7} = 1 ] [ 10m = 1 - \frac{40}{7} ] [ 10m = \frac{7}{7} - \frac{40}{7} ] [ 10m = \frac{-33}{7} ] [ m = \frac{-33}{70} ]

Finally, we find the correct values for ( m ) and ( w ) and then calculate the time for 6 men and 2 women:

[ 6m + 2w = \frac{6}{7} + \frac{2}{7} = \frac{8}{7} ]

The time taken is the reciprocal of this rate: [ \text{Time} = \frac{1}{\frac{8}{7}} = \frac{7}{8} \times 7 = 7 \times \frac{7}{8} = 14 ]

So the correct answer is:

{"correct_answer_key":"a"}

To solve this problem, we need to determine the work rates of both men and women and then use those rates to find out how long it will take for 6 men and 2 women to complete the same work.

First, let's determine the work rates:

1. 7 men and 6 women complete the work in 7 days.

   • Let ( M ) be the work rate of one man per day.
   • Let ( W ) be the work rate of one woman per day.
   • Total work done by 7 men and 6 women in one day is ( 7M + 6W ).
   • Since they complete the work in 7 days, the total work ( W_{total} ) can be expressed as: [ W_{total} = 7 \times (7M + 6W) ]

2. 5 men and 4 women complete the work in 10 days.

   • Total work done by 5 men and 4 women in one day is ( 5M + 4W ).
   • Since they complete the work in 10 days, the total work ( W_{total} ) can also be expressed as: [ W_{total} = 10 \times (5M + 4W) ]

Since both expressions represent the same total work, we can set them equal to each other: [ 7 \times (7M + 6W) = 10 \times (5M + 4W) ]

Expanding both sides: [ 49M + 42W = 50M + 40W ]

Rearranging to isolate ( M ) and ( W ): [ 49M + 42W - 50M - 40W = 0 ] [ -M + 2W = 0 ] [ M = 2W ]

Now we know the work rate of one man is twice the work rate of one woman. Let's substitute ( M = 2W ) back into one of the original equations to find the actual rates.

Using ( 7 \times (7M + 6W) = W_{total} ): [ 7 \times (7(2W) + 6W) = W_{total} ] [ 7 \times (14W + 6W) = W_{total} ] [ 7 \times 20W = W_{total} ] [ 140W = W_{total} ]

Now we know the total work ( W_{total} ) is ( 140W ).

Next, we need to determine how long it will take for 6 men and 2 women to complete the same work:

• The work rate of 6 men and 2 women is ( 6M + 2W ).
• Substituting ( M = 2W ): [ 6M + 2W = 6(2W) + 2W = 12W + 2W = 14W ]

The total work ( W_{total} ) is ( 140W ). To find the number of days ( D ) it takes for 14W of work per day to complete 140W of work: [ D = \frac{W_{total}}{\text{daily work rate}} = \frac{140W}{14W} = 10 ]

Therefore, it will take 6 men and 2 women 10 days to complete the work.

The correct answer is:

{"correct_answer_key":"e"}

To solve this problem, we need to determine the work rate of men and women and then use that to find out how long it will take for 6 men and 2 women to complete the work.

Let's denote:

• The work rate of one man per day as ( M )
• The work rate of one woman per day as ( W )

From the problem, we have two equations based on the given scenarios:

1. ( 7M + 6W = \frac{1}{7} ) (since 7 men and 6 women complete the work in 7 days)
2. ( 5M + 4W = \frac{1}{10} ) (since 5 men and 4 women complete the work in 10 days)

We can solve these equations to find ( M ) and ( W ).

Multiplying the first equation by 10 and the second equation by 7 to eliminate the fractions:

1. ( 70M + 60W = \frac{10}{7} )
2. ( 35M + 28W = \frac{7}{10} )

Now, let's multiply the second equation by 2 to align the coefficients of ( M ):

1. ( 70M + 60W = \frac{10}{7} )
2. ( 70M + 56W = \frac{14}{10} )

Subtract the second equation from the first: ( 4W = \frac{10}{7} - \frac{14}{10} ) ( 4W = \frac{100 - 98}{70} ) ( 4W = \frac{2}{70} ) ( W = \frac{1}{140} )

Substitute ( W ) back into one of the original equations to find ( M ): ( 7M + 6(\frac{1}{140}) = \frac{1}{7} ) ( 7M + \frac{6}{140} = \frac{1}{7} ) ( 7M + \frac{3}{70} = \frac{1}{7} ) ( 7M = \frac{1}{7} - \frac{3}{70} ) ( 7M = \frac{10 - 3}{70} ) ( 7M = \frac{7}{70} ) ( M = \frac{1}{70} )

Now, we need to find how many days ( D ) it will take for 6 men and 2 women to complete the work: ( 6M + 2W = \frac{1}{D} ) ( 6(\frac{1}{70}) + 2(\frac{1}{140}) = \frac{1}{D} ) ( \frac{6}{70} + \frac{2}{140} = \frac{1}{D} ) ( \frac{6}{70} + \frac{1}{70} = \frac{1}{D} ) ( \frac{7}{70} = \frac{1}{D} ) ( \frac{1}{10} = \frac{1}{D} ) ( D = 10 )

Thus, the correct answer is "e".

{"correct_answer_key":"e"}

คำอธิบายโดยย่อ:
กำหนดให้ชาย 1 คนทำงานได้ M ส่วนต่อวัน และหญิง 1 คนทำงานได้ W ส่วนต่อวัน
จากข้อมูล:
• 7M + 6W ทำงานเสร็จใน 7 วัน ⇒ (7M + 6W) × 7 = 1
• 5M + 4W ทำงานเสร็จใน 10 วัน ⇒ (5M + 4W) × 10 = 1

คำนวณได้ว่า M = 1/70 และ W = 1/140 เมื่อให้ชาย 6 คน หญิง 2 คน ทำงานร่วมกันจะได้อัตรางานต่อวันเท่ากับ 6M + 2W = 1/10 หมายความว่าต้องใช้เวลา 10 วันจึงจะเสร็จ

{"correct_answer_key":"e"}

7 ชาย + 6 หญิง ทำงานเสร็จใน 7 วัน
→ (7m + 6w)·7 = 1 ⇒ 7m + 6w = 1/7

5 ชาย + 4 หญิง ทำงานเสร็จใน 10 วัน
→ (5m + 4w)·10 = 1 ⇒ 5m + 4w = 1/10

แก้สมการคู่

7m + 6w = 1/7
5m + 4w = 1/10

ได้ w = 1/140, m = 1/70

อัตราทำงานต่อวันของ 6 ชาย + 2 หญิง
= 6(1/70) + 2(1/140)
= 6/70 + 1/70 = 7/70 = 1/10 งาน/วัน

จึงใช้เวลา t = 1 ÷ (1/10) = 10 วัน

{"correct_answer_key":"e"}

กำหนดให้กำลังการทำงานต่อวันของชาย = m และของหญิง = w

จากข้อมูล

1. 7 ชาย + 6 หญิง ทำงานเสร็จใน 7 วัน
   (7m + 6w) × 7 = งานทั้งหมด W

2. 5 ชาย + 4 หญิง ทำงานเสร็จใน 10 วัน
   (5m + 4w) × 10 = W

ตั้งสมการเท่ากัน
7(7m + 6w) = 10(5m + 4w)
49m + 42w = 50m + 40w
m = 2w ⇒ ชาย 1 คนทำงานได้เท่ากับหญิง 2 คน

ปริมาณงานทั้งหมด
W = 7 × (7m + 6w) = 7 × (7·2w + 6w) = 7 × 20w = 140w

ถ้ามี 6 ชาย + 2 หญิง
อัตราทำงานต่อวัน = 6m + 2w = 6·2w + 2w = 14w

จำนวนวันที่ใช้ = W / (อัตราต่อวัน) = 140w / 14w = 10 วัน

{"correct_answer_key":"e"}

ให้ m = อัตราการทำงานต่อวันของชาย 1 คน และ w = อัตราการทำงานต่อวันของหญิง 1 คน (งานทั้งหมด = 1)

1. 7 ชาย + 6 หญิง เสร็จใน 7 วัน
   (7m + 6w)·7 = 1 ⇒ 49m + 42w = 1 … (ก)

2. 5 ชาย + 4 หญิง เสร็จใน 10 วัน
   (5m + 4w)·10 = 1 ⇒ 50m + 40w = 1 … (ข)

ลบสมการ (ก) ออกจาก (ข):
(50m − 49m) + (40w − 42w) = 0 ⇒ m = 2w

แทน m = 2w ใน (ก):
49(2w) + 42w = 1 ⇒ 140w = 1 ⇒ w = 1/140
ดังนั้น m = 1/70

อัตรารวมของชาย 6 คน หญิง 2 คน:
6m + 2w = 6(1/70) + 2(1/140) = 6/70 + 1/70 = 7/70 = 1/10 งานต่อวัน

จึงใช้เวลา 10 วันจบงาน

{"correct_answer_key":"e"}

Let M be the work rate for a man and W for a woman. From the first scenario:
  7 men + 6 women finish the job in 7 days, so total work = 7*(7M + 6W).

From the second scenario:
  5 men + 4 women finish the job in 10 days, so total work = 10*(5M + 4W).

Since both equal the same work:
  7*(7M + 6W) = 10*(5M + 4W)
  49M + 42W = 50M + 40W
  Rearranging gives: 42W - 40W = 50M - 49M
  2W = M
  Thus, M = 2W.

Now substituting M = 2W into the first scenario total work:
  Total Work = 7*(7*(2W) + 6W) = 7*(14W + 6W) = 7*20W = 140W.

For 6 men and 2 women, the daily work rate is:
  6M + 2W = 6*(2W) + 2W = 12W + 2W = 14W per day.

Thus, the number of days required = Total Work / Daily Work Rate = 140W / 14W = 10 days.

Therefore, the correct answer is option E.

{"correct_answer_key": "e"}

We let m be the work rate of one man and w the work rate of one woman. From the first scenario, 7 men and 6 women working 7 days complete the work: (7m + 6w) × 7. From the second scenario, 5 men and 4 women working 10 days complete the same work: (5m + 4w) × 10. Setting these equal gives: 7(7m + 6w) = 10(5m + 4w) → 49m + 42w = 50m + 40w, which simplifies to m = 2w. Using this in the first equation, the total work equals 7(7(2w) + 6w) = 7(14w + 6w) = 7(20w) = 140w. Now, for 6 men and 2 women, the daily work rate is 6(2w) + 2w = 12w + 2w = 14w. Thus, the number of days required is total work/daily work = 140w/14w = 10 days.

{"correct_answer_key": "e"}

Let's denote the daily work rate of a man as m and that of a woman as w. From the first scenario, 7 men and 6 women complete the work in 7 days, so the total work (W) equals:   W = 7 × (7m + 6w).

From the second scenario, 5 men and 4 women complete the same work in 10 days:   W = 10 × (5m + 4w).

Setting these equal gives:   7(7m + 6w) = 10(5m + 4w)   49m + 42w = 50m + 40w   42w - 40w = 50m - 49m   2w = m  or  m = 2w.

Substitute m = 2w back into one of the equations to find the total work:   W = 7[7(2w) + 6w] = 7[14w + 6w] = 7 × 20w = 140w.

Now, for the team of 6 men and 2 women, their daily work rate is:   6m + 2w = 6(2w) + 2w = 12w + 2w = 14w.

The time required to complete the work is:   Time = Total Work / Daily Work Rate = 140w / 14w = 10 days.

Thus, the correct option is "10", which corresponds to option e.

{"correct_answer_key": "e"}

กำหนดให้ผลงานของชายต่อวัน = m และของหญิงต่อวัน = f
จากข้อมูล
(7m + 6f) × 7 = 1 หน่วยงาน
(5m + 4f) × 10 = 1 หน่วยงาน
แก้ระบบสมการจะได้ m = 1/70, f = 1/140
ดังนั้น ชาย 6 คนและหญิง 2 คน ทำงานได้วันละ (6×1/70 + 2×1/140) = 1/10 หน่วยงาน
จึงต้องใช้เวลา 10 วัน

{"correct_answer_key":"e"}

สมมติว่าผู้ชาย 1 คน ทำงานได้ m หน่วยต่อวัน ผู้หญิง 1 คน ทำงานได้ w หน่วยต่อวัน
จากข้อมูล
(7m + 6w)×7 = (5m + 4w)×10 ⇒ 49m+42w = 50m+40w ⇒ m=2w
งานทั้งหมด W = 49m+42w = 140w
ทีม 6 ชาย 2 หญิง ทำได้วันละ (6m+2w)=14w
ดังนั้นใช้เวลา = W ÷ วันละ = 140w ÷ 14w = 10 วัน

{"correct_answer_key":"e"}

จากสมการ
(7m+6w)·7 = W และ (5m+4w)·10 = W
แก้ได้ m = 2w, W = 140w
เมื่อใช้ 6 คนชาย และ 2 คนหญิง อัตรางานต่อวัน = 6m+2w = 14w
ดังนั้น วัน = W/(14w) = 140w/14w = 10

{"correct_answer_key":"e"}

To solve this problem, we need to determine the work rates of the men and women and then calculate how long it will take for 6 men and 2 women to complete the same work.

1. Calculate the total work in man-days and woman-days:

   • 7 men and 6 women complete the work in 7 days.
   • 5 men and 4 women complete the work in 10 days.

   Let's denote the work done by one man in one day as ( M ) and by one woman in one day as ( W ).

   From the first scenario: [ 7M \times 7 + 6W \times 7 = 49M + 42W ]

   From the second scenario: [ 5M \times 10 + 4W \times 10 = 50M + 40W ]

   Since both expressions represent the same total work, we equate them: [ 49M + 42W = 50M + 40W ]

   Simplifying gives: [ M = 2W ]

2. Substitute ( M = 2W ) into one of the equations to find the total work:

   Using the first scenario: [ 49(2W) + 42W = 98W + 42W = 140W ]

   So, the total work is ( 140W ).

3. Calculate the time for 6 men and 2 women:

   The work rate for 6 men and 2 women is: [ 6M + 2W = 6(2W) + 2W = 12W + 2W = 14W ]

   To find the number of days (( D )) it takes for them to complete the work: [ 14W \times D = 140W ]

   Solving for ( D ): [ D = \frac{140W}{14W} = 10 ]

Thus, it will take 6 men and 2 women 10 days to complete the work. The correct answer is option E.

{"correct_answer_key":"e"}

To solve this problem, let's break it down step by step:

Step 1: Define variables and equations

Let:

• The work rate of one man = $ M $ (work/day)
• The work rate of one woman = $ W $ (work/day)

The total amount of work is the same in both cases. We can express the total work as:

1. Case 1: $ 7M + 6W $ completes the work in 7 days: $$ \text{Work} = (7M + 6W) \times 7 = 49M + 42W $$

2. Case 2: $ 5M + 4W $ completes the work in 10 days: $$ \text{Work} = (5M + 4W) \times 10 = 50M + 40W $$

Since the total work is the same in both cases, we equate the two expressions: $$ 49M + 42W = 50M + 40W $$

Simplify: $$ M = 2W $$

This means one man works twice as fast as one woman.

Step 2: Calculate total work

Using $ M = 2W $, substitute into the total work equation from Case 1: $$ \text{Work} = 49M + 42W $$ Substitute $ M = 2W $: $$ \text{Work} = 49(2W) + 42W = 98W + 42W = 140W $$

Thus, the total work is $ 140W $.

Step 3: Calculate time for 6 men and 2 women

The combined work rate of 6 men and 2 women is: $$ 6M + 2W $$ Substitute $ M = 2W $: $$ 6M + 2W = 6(2W) + 2W = 12W + 2W = 14W $$

The total time required for 6 men and 2 women to complete the work is: $$ \text{Time} = \frac{\text{Total Work}}{\text{Work Rate}} = \frac{140W}{14W} = 10 , \text{days} $$

Final Answer:

The work will be completed in 10 days, which corresponds to option E.

{"correct_answer_key":"e"}

The problem involves determining the work rates of men (M) and women (W). From the given scenarios:

1. $7M + 6W = \frac{1}{7}$ (work per day)
2. $5M + 4W = \frac{1}{10}$ (work per day)

Solving these equations:

• Multiply the first equation by 10 and the second by 7 to eliminate fractions:
  • $70M + 60W = 10$
  • $35M + 28W = 7$
• Simplify to find $M = 2W$. Substituting back, we find $W = \frac{1}{140}$ and $M = \frac{1}{70}$.

For 6 men and 2 women:

• Total work rate: $6M + 2W = \frac{6}{70} + \frac{2}{140} = \frac{1}{10}$ (work per day).
• Time required: $1 / \frac{1}{10} = 10$ days.

{"correct_answer_key":"e"}

The problem involves determining the time required for 6 men and 2 women to complete a job, given two scenarios with different worker combinations. By setting up equations based on work rates, we solve for the individual rates of men and women.

Step-by-Step Explanation:

1. Let the work rate of one man be $ M $ (jobs/day) and one woman be $ W $ (jobs/day).
2. From the first scenario: $ 7M + 6W = \frac{1}{7} $ (since 7 men and 6 women complete 1 job in 7 days).
3. From the second scenario: $ 5M + 4W = \frac{1}{10} $ (since 5 men and 4 women complete 1 job in 10 days).
4. Solving the system of equations:
   • Multiply the first equation by 2: $ 14M + 12W = \frac{2}{7} $.
   • Multiply the second equation by 3: $ 15M + 12W = \frac{3}{10} $.
   • Subtract the first new equation from the second: $ M = \frac{1}{70} $.
   • Substitute $ M $ into the second original equation to find $ W = \frac{1}{140} $.
5. Calculate the combined rate for 6 men and 2 women: $ 6M + 2W = \frac{6}{70} + \frac{2}{140} = \frac{1}{10} $.
6. Time required: $ \frac{1}{\frac{1}{10}} = 10 $ days.

{"correct_answer_key":"e"}

To solve the problem, we first determine the work rates of men and women using the given scenarios:

1. Equation 1 (7 men + 6 women in 7 days):
   $ 7(7m + 6w) = 1 \Rightarrow 49m + 42w = 1 $
2. Equation 2 (5 men + 4 women in 10 days):
   $ 10(5m + 4w) = 1 \Rightarrow 50m + 40w = 1 $

Subtracting Equation 1 from Equation 2:
$ (50m + 40w) - (49m + 42w) = 0 \Rightarrow m = 2w $.

Substituting $ m = 2w $ into Equation 1:
$ 49(2w) + 42w = 1 \Rightarrow 140w = 1 \Rightarrow w = \frac{1}{140} $, so $ m = \frac{1}{70} $.

For 6 men and 2 women, their combined rate is:
$ 6 \cdot \frac{1}{70} + 2 \cdot \frac{1}{140} = \frac{6}{70} + \frac{2}{140} = \frac{14}{140} = \frac{1}{10} $ (job/day).

Thus, the time required to complete 1 job is $ \frac{1}{\frac{1}{10}} = 10 $ days.

{"correct_answer_key":"e"}

The problem involves determining the time required for 6 men and 2 women to complete a job, given the work rates of different groups. Let ( m ) be the work rate of a man (job/day) and ( w ) for a woman.

Equations from the problem:

1. ( (7m + 6w) \times 7 = 1 ) → ( 49m + 42w = 1 )
2. ( (5m + 4w) \times 10 = 1 ) → ( 50m + 40w = 1 )

Solving the system:

• Subtract Equation 1 from Equation 2:
  ( (50m - 49m) + (40w - 42w) = 0 ) → ( m - 2w = 0 ) → ( m = 2w ).
• Substitute ( m = 2w ) into Equation 1:
  ( 49(2w) + 42w = 1 ) → ( 140w = 1 ) → ( w = \frac{1}{140} ).
  Thus, ( m = \frac{1}{70} ).

Combined rate for 6 men and 2 women:
( 6m + 2w = 6\left(\frac{1}{70}\right) + 2\left(\frac{1}{140}\right) = \frac{6}{70} + \frac{1}{70} = \frac{7}{70} = \frac{1}{10} \text{ (job/day)} ).

Time required:
( \text{Time} = \frac{1}{\frac{1}{10}} = 10 \text{ days} ).

{"correct_answer_key":"e"}

เริ่มต้นด้วยการหาความสามารถในการทำงานของชายและหญิงคนหนึ่งในหนึ่งวัน

จากข้อมูลที่ให้มา:

1. ชาย 7 คน หญิง 6 คน ทำงานเสร็จใน 7 วัน
2. ชาย 5 คน หญิง 4 คน ทำงานเสร็จใน 10 วัน

เราสามารถตั้งสมการได้ดังนี้:

1. (7m + 6w) * 7 = 1 (งานทั้งหมด)
2. (5m + 4w) * 10 = 1 (งานทั้งหมด)

โดยที่ m คือความสามารถในการทำงานของชายคนหนึ่งในหนึ่งวัน และ w คือความสามารถในการทำงานของหญิงคนหนึ่งในหนึ่งวัน

จากสมการข้างต้น เราสามารถแก้หาค่า m และ w ได้ดังนี้:

1. 49m + 42w = 1
2. 50m + 40w = 1

ลบสมการที่ 2 ออกจากสมการที่ 1: -1m + 2w = 0 m = 2w

แทนค่า m ในสมการที่ 1: 49(2w) + 42w = 1 98w + 42w = 1 140w = 1 w = 1/140

แทนค่า w ในสมการ m = 2w: m = 2(1/140) = 1/70

ตอนนี้เรามีความสามารถในการทำงานของชายและหญิงคนหนึ่งในหนึ่งวันแล้ว เราสามารถหาว่าชาย 6 คน และหญิง 2 คน จะทำงานเสร็จภายในกี่วันได้

ชาย 6 คน และหญิง 2 คน จะทำงานได้ (6m + 2w) ในหนึ่งวัน = (6(1/70) + 2(1/140)) = (6/70 + 1/70) = 7/70 = 1/10

ดังนั้น ชาย 6 คน และหญิง 2 คน จะทำงานเสร็จภายใน 10 วัน

{"correct_answer_key":"e"}

{
  "correct_answer_key": "e"
}

Let $m$ be the amount of work a man can do in one day, and $w$ be the amount of work a woman can do in one day. From the first condition, 7 men and 6 women can complete the work in 7 days. So, the total work is $7(7m) + 6(7w) = 49m + 42w$. From the second condition, 5 men and 4 women can complete the work in 10 days. So, the total work is $5(10m) + 4(10w) = 50m + 40w$. Equating the two expressions for the total work, we have $49m + 42w = 50m + 40w$. Subtracting $49m + 40w$ from both sides, we get $2w = m$. Now, let $x$ be the number of days it takes for 6 men and 2 women to complete the work. The total work is $6(xm) + 2(xw) = 6xm + 2xw$. Since $m = 2w$, we can substitute $m$ with $2w$ in the equation: $6(x(2w)) + 2(xw) = 12xw + 2xw = 14xw$. We know that the total work is $49m + 42w = 49(2w) + 42w = 98w + 42w = 140w$. So, $14xw = 140w$. Dividing both sides by $14w$, we get $x = \frac{140w}{14w} = 10$. Therefore, it takes 10 days for 6 men and 2 women to complete the work.

Now, let's verify the work done in 7 days by 7 men and 6 women: Work done = $7 \times 7 \times m + 6 \times 7 \times w = 49m + 42w = 49(2w) + 42w = 98w + 42w = 140w$. Work done in 10 days by 5 men and 4 women: Work done = $5 \times 10 \times m + 4 \times 10 \times w = 50m + 40w = 50(2w) + 40w = 100w + 40w = 140w$. Work done in $x$ days by 6 men and 2 women: Work done = $6 \times x \times m + 2 \times x \times w = 6xm + 2xw = 6(2w)x + 2wx = 12wx + 2wx = 14wx$. We want $14wx = 140w$, so $14x = 140$, which means $x = 10$.

{"correct_answer_key":"e"}