A man a woman and a gurl can di work in 24 days 48 days and 144 days respectively how many girls must assist 3 men and 4 women to do the work in 3 days

To determine how many girls must assist 3 men and 4 women to complete the work in 3 days, we need to find the total work rate of the combined group and compare it to the required work rate.

First, let's calculate the work rates of a man, a woman, and a girl individually:

- A man's work rate: \( \frac{1 \text{ work unit}}{24 \text{ days}} = \frac{1}{24} \text{ work units per day} \)
- A woman's work rate: \( \frac{1 \text{ work unit}}{48 \text{ days}} = \frac{1}{48} \text{ work units per day} \)
- A girl's work rate: \( \frac{1 \text{ work unit}}{144 \text{ days}} = \frac{1}{144} \text{ work units per day} \)

Next, we find the combined work rate of 3 men and 4 women:

- Combined work rate of 3 men: \( 3 \times \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \text{ work units per day} \)
- Combined work rate of 4 women: \( 4 \times \frac{1}{48} = \frac{4}{48} = \frac{1}{12} \text{ work units per day} \)

Total combined work rate of 3 men and 4 women:
\[ \frac{1}{8} + \frac{1}{12} \]

To add these fractions, find a common denominator. The least common multiple of 8 and 12 is 24:

\[ \frac{1}{8} = \frac{3}{24} \]
\[ \frac{1}{12} = \frac{2}{24} \]

So,
\[ \frac{1}{8} + \frac{1}{12} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \text{ work units per day} \]

The total amount of work to be done is 1 work unit, and the time available is 3 days. Therefore, the required work rate is:
\[ \frac{1 \text{ work unit}}{3 \text{ days}} = \frac{1}{3} \text{ work units per day} \]

Let \( g \) be the number of girls needed. The work rate of \( g \) girls is:
\[ g \times \frac{1}{144} = \frac{g}{144} \text{ work units per day} \]

The total work rate of the combined group (3 men, 4 women, and \( g \) girls) must equal the required work rate:
\[ \frac{5}{24} + \frac{g}{144} = \frac{1}{3} \]

To solve for \( g \), first convert \( \frac{1}{3} \) to a denominator of 24 to combine with the other fractions:
\[ \frac{1}{3} = \frac{8}{24} \]

Thus,
\[ \frac{5}{24} + \frac{g}{144} = \frac{8}{24} \]

Subtract \( \frac{5}{24} \) from both sides:
\[ \frac{g}{144} = \frac{8}{24} - \frac{5}{24} = \frac{3}{24} \]

\[ \frac{g}{144} = \frac{3}{24} \]

To solve for \( g \), multiply both sides by 144:
\[ g = \frac{3}{24} \times 144 \]
\[ g = 3 \times 6 \]
\[ g = 18 \]

Therefore, 18 girls are needed to assist 3 men and 4 women to complete the work in 3 days.

Would you like more details or have any questions? Here are some related problems:

1. How long would it take for one man and one woman to complete the work together?
2. If the work rate of a girl doubled, how many girls would be needed to assist in completing the work in 3 days?
3. How many days would it take for 2 men, 3 women, and 4 girls to complete the work?
4. If only men were to complete the work, how many men would be needed to finish it in 3 days?
5. If the efficiency of each woman decreased by 25%, how would that affect the number of girls needed?
6. How many days would it take for 5 men to complete half of the work?
7. If one more woman joined the group of 3 men and 4 women, how many girls would be needed?
8. How many girls would be required if they worked only half the time as the men and women?

**Tip:** When solving work problems involving multiple workers, always find the individual work rates first and then sum them to find the combined work rate.

Explore how to determine the number of girls required to assist 3 men and 4 women in completing a task within 3 days. This problem involves calculating work rates, combining fractional work rates, and solving for the number of workers needed. Suitable for advanced high school students and beyond.

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