Understanding the Time and Work Problem

This question is a classic example of a time and work problem where the work rates of different individuals (men and boys) are related. We are given two scenarios where a certain amount of work is completed by different combinations of men and boys in different amounts of time. Our goal is to find the relationship between the work rate of a man and a boy and then use that relationship to determine the number of boys needed in a third scenario.

Establishing the Relationship Between Men and Boys' Work Rates

Let's denote the work rate of one man per day as $M$ and the work rate of one boy per day as $B$. The total work done is the product of the number of workers (adjusted for their individual rates) and the time taken.

According to the first part of the question:

• 5 men and 2 boys can do a certain amount of work in 30 days.
• The total work done by this group is $(5M + 2B) \times 30$.

Also, in the same timeframe:

• 7 men and 10 boys can do the same amount of work in 15 days.
• The total work done by this group is $(7M + 10B) \times 15$.

Since the work done is the same in both cases, we can set up the following equation:

$(5M + 2B) \times 30 = (7M + 10B) \times 15$

Now, let's solve this equation to find the relationship between $M$ and $B$:

$(5M + 2B) \times 2 = (7M + 10B) \times 1 \quad \text{(Dividing both sides by 15)}$
$10M + 4B = 7M + 10B$

Rearranging the terms to group $M$ and $B$:

$10M - 7M = 10B - 4B$
$3M = 6B$

Dividing both sides by 3:

$M = 2B$

This tells us that the work rate of one man is equivalent to the work rate of two boys.

Calculating the Total Work

Now that we know the relationship between $M$ and $B$, we can calculate the total amount of work. We can use either of the initial scenarios. Let's use the first one (5 men and 2 boys in 30 days) and express the total work in terms of boy-days ($B$).

Total Work $= (5M + 2B) \times 30$

Substitute $M = 2B$ into the equation:

Total Work $= (5 \times (2B) + 2B) \times 30$
Total Work $= (10B + 2B) \times 30$
Total Work $= (12B) \times 30$
Total Work $= 360B$

So, the total work is equivalent to 360 boy-days.

Determining the Number of Boys Required

The question asks how many boys should join 40 men to do the same work (360B) in 4 days. Let the number of boys needed be $x$.

The group consists of 40 men and $x$ boys. Their combined work rate per day is $(40M + xB)$.

They need to complete the total work (360B) in 4 days. So, the total work done by this group in 4 days is:

$(40M + xB) \times 4$

We know that $M = 2B$. Substitute this into the equation:

Total Work $= (40 \times (2B) + xB) \times 4$
Total Work $= (80B + xB) \times 4$
Total Work $= (80 + x)B \times 4$

This total work must be equal to 360B:

$(80 + x)B \times 4 = 360B$

Since $B$ represents a work rate, it's a non-zero value. We can divide both sides by $B$:

$(80 + x) \times 4 = 360$

Now, solve for $x$:

$80 + x = \frac{360}{4}$
$80 + x = 90$
$x = 90 - 80$
$x = 10$

Therefore, 10 boys should join the 40 men to complete the work in 4 days.

Summary of Steps

1. Define variables for the work rates of men ($M$) and boys ($B$).
2. Set up an equation based on the first two scenarios where the work done is equal.
3. Solve the equation to find the relationship between $M$ and $B$.
4. Calculate the total amount of work using the relationship found and one of the initial scenarios.
5. Set up an equation for the third scenario (40 men and $x$ boys in 4 days) and equate it to the total work.
6. Solve the final equation for $x$ to find the number of boys needed.

Revision Table: Key Concepts in Time and Work Problems

Additional Information on Time and Work Calculations

Time and work problems often involve inverse proportionality. If a group of workers increases, the time required to complete the same amount of work decreases, assuming their individual work rates remain constant. Conversely, if the work to be done increases, the time required will also increase for the same group of workers.

Problems involving different types of workers (like men and boys) require you to first establish a common unit of work or a relationship between their efficiencies, as we did by finding $M = 2B$. Once this relationship is known, the problem simplifies into a standard time and work calculation.

Always ensure you are consistent with the units. If work rates are per day, time should be in days. The total work will then be in units of 'worker-days' or similar.

For solving such problems efficiently during exams, practice establishing the work rate relationship quickly and converting all workers into equivalent units of the worker with the base efficiency (usually the one assumed to be less efficient, like boys in this case).