Solving Work Problems with Fractions and Ratios: A Step-by-Step Guide

Solving work problems often requires a clear understanding of fractions, ratios, and combined work rates. These types of problems are common in mathematics, especially in standardized tests and practical applications. Here, we will walk through a few examples to illustrate the techniques needed to solve such problems effectively and efficiently.

Sample Problem 1

A and B together can complete a work in 4 days. They started together but after 2 days B left the work. If the work is completed after 2 more days, in how many days could B alone do the work?

Solution:

First, we find the fraction of work A and B together can complete in one day.

A B's combined work rate (frac{1}{4}) of the work per day.

In 2 days, A and B together complete:

2 days * (frac{1}{4}) (frac{1}{2}) of the work.

After B leaves, A completes the remaining (frac{1}{2}) of the work in 2 more days. Thus, A's work rate is:

(frac{1}{2}) of the work in 2 days (frac{1}{4}) of the work per day.

Now, we find B's work rate. We know A and B together complete (frac{1}{4}) of the work per day, and A completes (frac{1}{4}) of the work per day alone.

B's work rate (frac{1}{4} - frac{1}{4} 0).

Necessary correction: Let's re-evaluate the work rates correctly.

Let A's work rate be (frac{1}{a}) and B's work rate be (frac{1}{b}).

A B's combined work rate: (frac{1}{a} frac{1}{b} frac{1}{4}).

2 days * (frac{1}{4}) (frac{1}{2}) of the work. Remaining: (frac{1}{2}).

A completes the remaining (frac{1}{2}) of the work in 2 days.

A's work rate (frac{1}{a} frac{1}{6}).

Thus, B's work rate (frac{1}{4} - frac{1}{6} frac{1}{12}).

B alone can do the work in (12) days.

Answer: (12) days.

Sample Problem 2

A in 1 day completes (frac{1}{10})th of the work; B in 1 day completes (frac{1}{15})th of the work. Together in 1 day, they complete (frac{5}{30} frac{1}{6}) of the work. In 2 days they have completed (frac{1}{3}) of the work. If A leaves after 2 days, B alone will complete the remaining (frac{2}{3}) of the work in:

B's work rate: (frac{1}{b} frac{2}{3} / 10 frac{2}{30} frac{1}{15}).

B will complete the remaining (frac{2}{3}) of the work in 15 days, but since B's work rate is (frac{1}{15}) per day, it takes 15 days.

Answer: 15 days.

Sample Problem 3

A does half the job in 3 days, so A's work rate is (frac{1}{2} / 3 frac{1}{6}) of the work per day. B would complete the remaining (frac{1}{2}) in the same 4 days, so B's work rate is (frac{1}{8}) of the work per day. B takes 8 days to complete the remaining (frac{1}{2}) of the job.

Answer: 8 days.

Sample Problem 4

We know that A and B together can complete the work in 4 days, so their combined work rate is (frac{1}{4}).

A can do half the job in 3 days, so A's work rate is (frac{1}{2} / 3 frac{1}{6}).

B's work rate: (frac{1}{4} - frac{1}{6} frac{3}{12} - frac{2}{12} frac{1}{12}).

B will need 12 days to complete the entire job alone.

Answer: 12 days.

Sample Problem 5

Assume the job is making 60 widgets. A and B together make 60 widgets in 4 days, so their combined work rate is 15 widgets per day. A can make 30 widgets in 3 days, so A's work rate is 10 widgets per day.

B's work rate is 10 - 5 5 widgets per day. B will need 60 / 5 12 days to complete the job alone.

Answer: 12 days.

Summary

When solving work problems, convert everything to fractions of the job done per day. Use the combined work rate to determine the individual work rates. This method simplifies complex problems and ensures accurate solutions. Understanding these techniques will enhance your ability to tackle similar problems, making it an essential skill for both academic and practical scenarios.