Ratios & Percents — Core Skills

Ratios and Proportions: The Basics

Part 1 of 2 — The Basics

A ratio compares two amounts. If a bag has $6$ red marbles and $9$ blue marbles, the ratio of red to blue is $6 : 9$.

You simplify a ratio the same way you simplify a fraction: divide both numbers by the same value. Both $6$ and $9$ divide by $3$, so $6 : 9$ becomes $2 : 3$.

A proportion says two ratios are equal, like $\frac{2}{4} = \frac{6}{12}$.

Solving a proportion by scaling

When one side of a proportion is a whole-number multiple of the other, you can scale straight across.

Worked example. A recipe uses $3$ cups of flour for $6$ servings. How many cups are needed for $18$ servings?

Step 1. Write the proportion. Keep cups on top and servings on the bottom on both sides:

$\frac{3}{6} = \frac{x}{18}$

Step 2. Ask how the bottoms are related. From $6$ to $18$, you multiply by $3$, because $6 \times 3 = 18$.

Step 3. Do the same thing to the top. $3 \times 3 = 9$.

So $x = 9$. The recipe needs $9$ cups of flour.

Step 4 (the check). Does $\frac{3}{6}$ equal $\frac{9}{18}$? Both reduce to $\frac{1}{2}$, so yes.

The one thing to be careful about

Whatever is on top on the left side must be on top on the right side too. If you put cups over servings on one side, do the same on the other side. Flipping one side is the most common mistake on these questions.

Percents: Practice

Part 2 of 2 — Practice

Percent means "out of one hundred." So $25\%$ means $25$ out of $100$, which is the same as the decimal $0.25$.

The steps for "what is X percent of Y"

1. Turn the percent into a decimal by moving the decimal point two places to the left. $25\%$ becomes $0.25$. $60\%$ becomes $0.60$. $8\%$ becomes $0.08$.
2. The word of means multiply.
3. Multiply the decimal by the number.

Worked example. What is $20\%$ of $60$?

$20\%$ becomes $0.20$. Then $0.20 \times 60 = 12$. So $20\%$ of $60$ is $12$.

Two shortcuts worth memorizing

• $50\%$ of a number is half of it. Divide by $2$.
• $10\%$ of a number moves the decimal point one place left. $10\%$ of $70$ is $7$.

Discounts take one extra step

A discount tells you how much comes off. The price you pay is what is left.

1. Find the amount of the discount.
2. Subtract that amount from the original price.

Worked example. A jacket costs \50$and is$10%$off. First,$10%$of$50$is$5$. Then subtract:$50 - 5 = 45$. The sale price is$$45$.