Proportion problems are word problems where the items in the question are proportional to each other. In these lessons, we will learn the two main types of proportional problems: Directly Proportional Problems and Inversely Proportional Problems.

**Related Pages:**

Proportions

Direct Variations

More Algebra Lessons

The following diagrams show the formulas and graphs for directly proportional and inversely proportional problems. Scroll down the page for examples and solutions.

There are many situations in our daily lives that involve direct proportion.

For example, a worker may be paid according to the number of hours he worked. The two quantities, the number of hours worked (x) and the amount paid (y), are related in such a way that when x changes, y changes proportionately.

In general, when two variables x and y are such that the ratio \(\frac{y}{x}\)
remains a constant, we say that y is **directly proportional** to x.

If we represent the constant by k, then we can get the equation:

\(\frac{y}{x}\) = k or y = kx where k ≠ 0.

In notation, direct proportion is written as

y ∝ x

**Example 1:**

If y is directly proportional to x and given y = 9 when x = 5, find:

a) the value of y when x = 15

b) the value of x when y = 6

**Solution:**

a) Using the fact that the ratios are constant, we get

\(\frac{9}{5}\) = \(\frac{y}{15}\)

⇒ y = \(\frac{9}{5}\) × 15

⇒ y = 27

b) \(\frac{9}{5}\) = \(\frac{6}{x}\)

⇒ x = \(\frac{5}{9}\) × 6

⇒ x = \(\frac{10}{3}\) = \(3\frac{1}{3}\)

**Example 2:**

Jane ran 100 meters in 15 seconds. How long did she take to run 2 meter?

**Solution:**

\(\frac{100}{15}\) = \(\frac{y}{2}\)

⇒ y = \(\frac{15}{100}\) × 2

⇒ y = 0.3

Answer: She took 0.3 seconds.

**Example 3:**

A car travels 125 miles in 3 hours. How far would it travel in 5 hours?

**Solution:**

\(\frac{125}{3}\) = \(\frac{y}{5}\)

⇒ y = \(\frac{125}{3}\) × 5

⇒ y = \(208\frac{1}{3}\)

Answer: He traveled \(208\frac{1}{3}\) miles.

**Examples:**

Use proportions to find the missing value

- 8 inches in 25 minutes ; 28 inches in
*x*minutes - 3 gallons in 7 hours ;
*x*gallons in 20 hours

**Example:**

Arthur is typing a paper that is 390 words long. He can type 30 words in a minute. How long
will it take for him to type the paper?

There are also many situations in our daily lives that involve inverse proportion.

For example, the number of days required to build a bridge is inversely proportional to the number of workers. As the number of workers increases, the number of days required to build would decrease.

The two quantities, the number of workers (x) and the number of days required (y), are related in such a way that when one quantity increases, the other decreases.

In general, when two variables x and y are such that

xy = k where k is a non-zero constant, we say that y is **inversely proportional** to x.

In notation, inverse proportion is written as

y ∝ \(\frac{1}{x}\)

**Example:**

Suppose that y is inversely proportional to x and that y = 8 when x = 3. Calculate the value
of y when x = 10.

**Solution:**

Using the fact that the products are constant, we get

3 × 8 = 10y

⇒ y = \(\frac{24}{10}\) = \(2\frac{2}{5}\)

**Example:**

It takes 4 men 6 hours to repair a road. How long will it take 7 men to do the job if they
work at the same rate?

**Solution:**

4 × 6 = 7y

⇒ y = \(\frac{24}{7}\) = \(3\frac{3}{7}\)

Answer: They will take \(3\frac{3}{7}\) hours.

**How To Solve A Word Problem That Involves Inverse Proportion**

**Examples:**

- Suppose x and y are inversely proportional. If x = 24 and y = 18, then what is x when y = 90?
- It will take 30 hours for 8 graders to grade all the USAMTS papers. If the graders all grade at the same rate, then how many graders do we need to get the grading done in 12 hours?
- Ohm’s Law states that current and resistance are inversely proportional. The resistance of a wire is also directly proportional to the length of the wire. If I wish to double the current flowing through a section of a circuit, and all I can change is the length of the wire, then how should I alter the length of the wire?

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