Solving and Verifying One-Variable Proportions

Hello. Last time, you represented proportional relationships as equivalent ratios by keeping the quantities in the same order and scaling both parts by the same factor. Now you will use that idea when one of the four values is missing.

By the end of this lesson, you should be able to set up a one-variable proportion, solve it using cross multiplication, and verify the result by substitution. This is a useful skill for the similar-polygon work that comes next, because corresponding side lengths form proportions.

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A proportion is an equation with one missing value

A proportion says that two ratios have the same value:

$$\frac{a}{b}=\frac{c}{d}$$

When one number is unknown, use a variable such as $x$. Your task is to find the value of $x$ that makes the ratios truly equivalent.

For example:

$$\frac{3}{5}=\frac{x}{20}$$

The fraction on the right must have the same value as $\frac{3}{5}$. Since $5$ was multiplied by $4$ to make $20$, you could use the equivalent-ratio idea from the last lesson: $3$ must also be multiplied by $4$. Therefore, $x=12$.

That scaling method is fast when the multiplier is obvious. Cross multiplication is a reliable general method, especially when the scale factor is not easy to see.

Before doing any calculation in a word problem, check the ratio order. If the first ratio is books over days, the second must also be books over days. A correct calculation cannot fix a proportion whose quantities have been placed in a different order.

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The cross-multiplication method

For a proportion

$$\frac{a}{b}=\frac{c}{d}$$

with nonzero denominators, the cross products are equal:

$$ad=bc$$

In other words, multiply the top of the left fraction by the bottom of the right fraction. Then multiply the bottom of the left fraction by the top of the right fraction. Set those two products equal.

This is not a trick to memorize without meaning. Since the two fractions are equal, multiplying both sides by $bd$ clears both denominators:

$$\frac{a}{b}=\frac{c}{d}$$ $$ad=bc$$

After that, solve the resulting equation as usual.

Consider this proportion:

$$\frac{x}{15}=\frac{1}{3}$$

Cross multiply:

$$3x=15\cdot 1$$

Simplify:

$$3x=15$$

Divide both sides by $3$:

$$x=5$$

The diagonal lines in the image are a memory aid, but write the multiplication equation clearly on your paper. Showing

$$3x=15$$

makes your reasoning visible and gives you a line where you can catch arithmetic mistakes.

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Verification: substitute, simplify, compare

Solving produces a possible value. Verifying tests whether that value really makes the original proportion true.

For the example above, you found:

$$x=5$$

Return to the original equation:

$$\frac{x}{15}=\frac{1}{3}$$

Substitute $5$ for $x$:

$$\frac{5}{15}=\frac{1}{3}$$

Simplify the left side:

$$\frac{1}{3}=\frac{1}{3}$$

The sides match, so $x=5$ is verified.

Use this routine every time:

1. Write the original proportion again.
2. Replace the variable with your answer.
3. Simplify each ratio, or convert each to decimals if that is easier.
4. Confirm that the two values are equal.
5. In a word problem, make a quick reasonableness check: does the size and unit of the answer fit the situation?

A substitution check is particularly helpful because it catches many slips: multiplying the wrong numbers, dividing incorrectly, or copying a value inaccurately.

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When the variable is in the denominator

Cross multiplication works no matter where $x$ appears. For example:

$$\frac{7}{x}=\frac{21}{30}$$

Multiply the diagonal pairs:

$$7\cdot 30=21x$$ $$210=21x$$

Divide both sides by $21$:

$$10=x$$

Now verify it by substitution:

$$\frac{7}{10}=\frac{21}{30}$$

Simplify the right-hand fraction by dividing its numerator and denominator by $3$:

$$\frac{7}{10}=\frac{7}{10}$$

So the solution is correct:

$$x=10$$

Notice the difference between solving and checking. Cross multiplication helped you find $x$; substitution confirmed that the answer makes the original ratios equal.

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A clean quiz-paper format

For most one-variable proportions, your written work can follow this format:

$$\frac{\text{first ratio}}{\text{second ratio}}=\frac{\text{matching ratio}}{\text{matching ratio}}$$ $$\text{cross product}=\text{cross product}$$ $$\text{simplify and isolate }x$$ $$\text{substitute the value of }x\text{ into the original proportion}$$

You do not need to write extra words for every arithmetic operation, but do keep these important lines visible:

• the original proportion;
• the cross-product equation;
• the final value of $x$;
• the substitution check.

That layout helps both with accuracy and with earning method marks if a small arithmetic error occurs.

For a short retrieve-check-correct study cycle, cover the worked examples in this lesson. On blank paper, reconstruct the method from memory: write the cross-product rule, list the verification routine, then uncover the lesson and correct only the parts you missed. Keep a note of whether the miss was a setup, calculation, or checking issue.

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A one-variable proportion is solved by preserving matching ratio order, setting the cross products equal, and isolating the variable. The solution is not complete until you substitute it back into the original proportion and show that both ratios are equal.

Next, you will apply this proportional reasoning to similar polygons by identifying which vertices and sides correspond.