We want to find the number of boys needed to plow 45 acres of land in 6 days, given that 7 men are already available. Let's call the number of boys we need B.

We know that 5 men and 3 boys can plow 23 acres in 4 days. That means that the total number of worker-days is:

5 men * 4 days + 3 boys * 4 days = 20 men-days + 12 boy-days = 32 worker-days

We can use this information to set up a proportion:

(20 men-days + 12 boy-days) / 23 acres = (7 men-days + B boy-days) / 45 acres

Simplifying this equation, we get:

(20 + 12B) / 23 = (7 + B) / 45

Cross-multiplying and simplifying, we get:

900 + 540B = 23(7 + B)

900 + 540B = 161 + 23B

517B = 739

B = 1.43 (rounded to two decimal places)

Since we cannot have a fractional number of workers, we need to round up to the nearest whole number. Therefore, we need 2 boys to plow 45 acres of land in 6 days with 7 men.

other solution

If one man can plow one acre of land in one day, then in 4 days, 5 men can plow:

5 men * 4 days * 1 acre per day per man = 20 acres

Similarly, in 4 days, 3 boys can plow:

3 boys * 4 days * 3/5 acres per day per boy = 7.2 acres (rounded to one decimal place)

So, in 4 days, 5 men and 3 boys together can plow:

20 acres + 7.2 acres = 27.2 acres (rounded to one decimal place)

Using this information, we can set up a proportion to find how many workers are needed to plow 45 acres of land in 6 days. If we assume that each worker has the same plowing rate as before, then we have:

(5 men + 3 boys) / 27.2 acres = (7 men + B boys) / 45 acres

Simplifying this equation, we get:

(5 + 3B/5) / 27.2 = (7 + B) / 45

Cross-multiplying and simplifying, we get:

225 + 135B = 338.4 + 45B

90B = 113.4

B = 1.26 (rounded to two decimal places)

Since we cannot have a fractional number of workers, we need to round up to the nearest whole number. Therefore, we need 2 boys to plow 45 acres of land in 6 days with 7 men.