SOLVING UNKNOWNS IN PROPORTION

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Proportion, as I told you before, states that two ratios are the same.

So, the statement $\frac{7}{4}=\frac{28}{16}$ because the ratios are the same when simplify.

A simpler way of determining if two ratios are proportional is to use cross-multiplication. This is because two ratios are proportional if their cross-products are the same.

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

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

$$112=112$$

Because their cross-products are the same, the two ratios are the same

But, what if we replace 7 with x so that it becomes: 

$$\frac{x}{4}=\frac{28}{14}$$

Since we know that cross product of a proportion is the same.

$$x \times 14= 28  \times 4$$

$$14x=112$$

$$x=7$$

Using this fact, let's solve few examples

Example 1

Solve for x in $\frac{145}{232}=\frac{10}{2y}$

Solution:

By cross multiplication

$$145 \times 2y=232 \times 10$$

$$290y=2320$$

$$\frac{290y}{290}=\frac{2320}{290}$$

$$y=8$$

Example 2

Solve $\frac{8}{10z}=\frac{0.4}{1.4}$

Solution:

By cross multiplication

$$8 \times 1.4= 10z \times 0.4$$

$$11.2=4z$$

$$z=2.8$$

Example 3

Solve for x in $\frac{5x}{150}=\frac{\frac{2}{3}}{5}$

Solution:

First, we evaluate the right-hand side

$$\frac{5x}{150}=\frac{2}{3}\times\frac{1}{5}$$

$$\frac{5x}{150}=\frac{2}{15}$$

$$5x ×15=150 × 2$$

$$75x=300$$

$$x=4$$

Example 4

Solve for x in $\frac{32}{100}=\frac{40}{4y}$

Solution:

By cross multiplication

$$128y=4000$$

$$\frac{128y}{128}=\frac{4000}{128}$$

$$y=31.25$$

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