PSAT 8/9 Math · Lesson 3 of 4
Ratios, percentages, and data
Keep the order of a ratio, choose the correct whole for a percentage, and read a table before you calculate.
Scale both parts of a ratio
A ratio 2:5 means 2 parts of the first quantity for every 5 parts of the second. If the second quantity becomes 15, it has been multiplied by 3, so the first quantity is also multiplied by 3. The new ratio is 6:15, which still simplifies to 2:5.
Do not add the ratio numbers and call that the total unless the question asks for the whole mixture. Concentrate and water are different parts. Their sum is a third quantity.
Percent of which amount?
A percentage needs a reference amount. Twenty percent of 80 is 0.20 × 80 = 16. Increasing 80 by 20% gives 80 + 16 = 96, which is the same as multiplying by 1.20.
In a table, the phrase “of all students” and “of students in grade 8” use different denominators. Circle the group the question restricts before you divide. For a distribution, the mean uses every value, while the median depends on the ordered middle.
Worked example
A shirt priced at $40 is reduced by 15%. What is the sale price?
- Fifteen percent of 40 is 0.15 × 40 = 6.
- Subtract the discount: 40 − 6 = 34.
- Equivalently, the buyer pays 85% of the price: 0.85 × 40 = 34.
Why this works. The discount is not subtracted from 15. It is 15% of the original price.
Check your understanding
- Multiply both parts of a ratio by the same factor.
- Name the reference amount before calculating a percentage.
- Identify whether a data question asks for a total, a rate, a mean, or a restricted group.