6.5 Solve Proportions and their Applications

Understanding Proportions

A proportion is an equation that says two ratios are equal. Since percents are really just ratios out of 100, proportions give you a reliable method for solving percent problems. They also show up in scale drawings, unit pricing, and similar figures.

Definition and Identification of Proportions

A proportion is an equation stating that two ratios are equivalent. It takes the form:

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

You can also write proportions with colon notation: $2:3 = 4:6$. Both formats mean the same thing.

The key property of a proportion is that the cross products are equal. If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$. This gives you a quick way to check whether two ratios actually form a proportion.

For example, do $\frac{3}{4}$ and $\frac{9}{12}$ form a proportion?

• Cross multiply: $3 \times 12 = 36$ and $4 \times 9 = 36$
• The cross products are equal, so yes, it's a proportion.

Methods for Solving Proportions

When a proportion has an unknown variable, cross multiplication turns it into a simple equation you can solve. Here's the process:

1. Write the proportion. For example: $\frac{2}{x} = \frac{6}{9}$
2. Cross multiply. Multiply the numerator of each fraction by the denominator of the other: $2 \times 9 = x \times 6$
3. Simplify. This gives you $18 = 6x$
4. Solve for the variable. Divide both sides by 6: $x = 3$
5. Check your answer. Substitute back in: $\frac{2}{3} = \frac{6}{9}$. Cross products: $2 \times 9 = 18$ and $3 \times 6 = 18$. It checks out.

Always verify by plugging your answer back into the original proportion. This catches arithmetic mistakes before they cost you points.

Applying Proportions

Real-World Applications of Proportions

Proportions come up whenever two quantities have a consistent ratio:

• Scale drawings and maps: If 1 inch on a map represents 50 miles, you can set up $\frac{1}{50} = \frac{\text{map distance}}{\text{actual distance}}$ to find unknown distances.
• Unit pricing: If 3 items cost $12, what do 7 items cost? Set up $\frac{3}{12} = \frac{7}{x}$, cross multiply to get $3x = 84$, and solve: $x = 28$.
• Similar triangles: Corresponding sides of similar triangles are proportional. If one triangle has sides of 3 and 5, and the matching side on a similar triangle is 9, you can find the unknown side with $\frac{3}{5} = \frac{9}{x}$.
• Direct proportionality: When $y$ is directly proportional to $x$, the ratio $\frac{y}{x}$ stays constant, so $\frac{y_1}{x_1} = \frac{y_2}{x_2}$.

Converting Percent Problems to Proportions

A percent is just a ratio out of 100. That means every percent problem can be written as a proportion:

$\frac{\text{percent}}{100} = \frac{\text{part}}{\text{whole}}$

For example, "25% of what number is 30?" becomes:

$\frac{25}{100} = \frac{30}{x}$

Cross multiply: $25x = 3000$, so $x = 120$.

You can also convert the percent to a decimal ($0.25x = 30$) or a fraction ($\frac{1}{4}x = 30$) and solve that way. All three approaches give the same answer. The proportion method is especially useful because it keeps the setup consistent no matter what's missing.

Interpreting Percentage Word Problems

The trickiest part of percent word problems is figuring out which number is the part and which is the whole. Here's how to break them down:

1. Identify the percent. This goes over 100.
2. Identify the whole. This is the total amount (often follows the word "of").
3. Identify the part. This is the portion of the whole (often follows the word "is").
4. Set up the proportion: $\frac{\text{percent}}{100} = \frac{\text{part}}{\text{whole}}$
5. Solve and check that your answer makes sense. If the percent is less than 100, the part should be smaller than the whole.

For example: "40% of a number is 30." The percent is 40, the part is 30, and the whole is unknown.

$\frac{40}{100} = \frac{30}{x}$

Cross multiply: $40x = 3000$, so $x = 75$. Quick check: 40% means less than half, and 30 is less than half of 75. That makes sense.

Problem-Solving with Proportions

Setting Up and Solving Proportion Equations

For any proportion word problem, follow these steps:

1. Read the problem carefully and identify what you know and what you need to find.
2. Assign a variable to the unknown quantity.
3. Write two ratios that should be equal. Make sure matching units are in the same position (numerator with numerator, denominator with denominator).
4. Cross multiply and solve the resulting equation.
5. Check your answer in two ways: substitute it back into the proportion, and ask whether it makes sense in context.

A common mistake is mixing up which values go where. To avoid this, label your ratios. For instance, if the problem says "5 out of every 8 students prefer pizza, and there are 200 students total," you'd write:

$\frac{5}{8} = \frac{x}{200}$

Both numerators represent students who prefer pizza, and both denominators represent total students. Keeping your labels consistent prevents setup errors.