SOLVING PROPORTION

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A proportion states that two ratios or fractions are equal. For example, The statement $\frac{7}{4}=\frac{28}{16}$ is a proportion because the right-hand side is the same as the left hand when simplified.

To check if two ratios are proportional, we use cross multiplications. A proportion, say $\frac{7}{4}=\frac{28}{16}$, is true if the cross products are equal and cross products are usually obtained from cross-multiplication.

Now, let's check if $\frac{7}{4}=\frac{28}{16}$ is a valid proportion.

$$\frac{7}{4}=\frac{28}{16}$$

Now, let's cross multiply

$$7 \times 16 =28 \times 4$$

$$112=112$$

Because the cross products are the same, $\frac{7}{4}=\frac{28}{16}$ is a proportion

Example 1

Is $\frac{14}{18}=\frac{280}{360}$ a proportion or not?

Solution:

Recall that two ratios are proportional if, and if only, the cross-products are the same.

 $$\frac{14}{18}=\frac{280}{360}$$

By cross-multiplication

$$14 \times 360=18 \times 280$$

$$5040=5040$$

Because the cross products are the same, the two ratios are proportional

Example 2

Is $\frac{8.8}{10}=\frac{2}{5}$ a proportion or not?

Solution:

Recall that two ratios are proportional if the cross-products are the same.

 $$\frac{8.8}{10}=\frac{2}{5}$$

By cross-multiplication

$$8.8 \times 5=10 \times 2$$

$$44=20$$

Because the cross products are not the same, the two ratios are not proportional

Example 3

Is $\frac{3}{2.5}=\frac{156}{130}$ a proportion or not?

Solution:

$$\frac{3}{2.5}=\frac{156}{130}$$

By cross multiplication 

$$3 \times 130= 2.5 \times 156$$

$$390=390$$

Because the cross products are the same, the two ratios are a proportional


Proportion can be used to solve some equality problems. Once we have three of the four numbers that made up a proportion, we can find the unknown number

Example 4

Solve for x in $\frac{2x}{12}=\frac{2}{3}$ 

Solution:

$$\frac{2x}{12}=\frac{2}{3}$$

Because the cross-product of proportion is equal, we cross multiply.

$$2x \times 3= 12 \times 2$$

$$6x= 24$$

$$x=4$$

To check your answer, insert 4 for x in the original expression

$$\frac{2(4)}{12}=\frac{2}{3}$$ 

$$\frac{8}{12}=\frac{2}{3}$$ 

Cross multiplying, the result is

$$24=24$$.. which means the answer is correct.

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Example 5

If $\frac{36}{135}=\frac{8}{x}$, Determine the value of x

Solution:

By cross-multiplication

$$36 \times x=135 \times 8$$

$$36x=1080$$

$$\frac{36x}{36}=\frac{1080}{36}$$

$$x=30$$

There you have it!. Got questions that you want us to help you understand? Ask us here

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