PSAT 10 Percentages, Ratios & Proportions Cheat Sheet

1) What You Need to Know

Percent, ratio, and proportion questions show up constantly on the PSAT 10 because they test whether you can translate words into relationships and then compute accurately. Most problems are about part–whole, comparison, or change.

Core ideas (in one place)

• Percent means “per 100.”
  $p\% = \frac{p}{100}$
• Ratio compares quantities by division.
  $\text{ratio } a:b = \frac{a}{b}$
• Proportion is an equation stating two ratios are equal.
  $\frac{a}{b} = \frac{c}{d}$

When you use what

• Use percent for: discounts, tax, tips, percent change, interest, percent of a number.
• Use ratios for: part-to-part or part-to-whole comparisons, mixing, splitting, scaling.
• Use proportions for: constant rate problems, scale drawings, similar-figure comparisons, “for every,” and “direct variation.”

Critical reminder: Always identify what’s the whole and what’s being compared before calculating.

2) Step-by-Step Breakdown

A) “Percent of” / “What percent” / “Percent change” problems

Universal translation:
$\text{Percent} \times \text{Whole} = \text{Part}$
Where percent is written as a decimal.

1) Find a percent of a number

1. Convert percent to decimal.
2. Multiply by the whole.

Example: Find $35\%$ of $240$.
$0.35 \times 240 = 84$

2) Find what percent one number is of another

1. Identify part and whole.
2. Compute $\frac{\text{part}}{\text{whole}}$.
3. Convert to percent (multiply by $100$).

Example: $18$ is what percent of $60$?
$\frac{18}{60} = 0.3$
$0.3 = 30\%$

3) Percent increase/decrease

1. Compute change: $\text{new} - \text{old}$.
2. Divide by original (old).
3. Convert to percent.

Example: Price goes from $50$ to $65$. Percent increase?
$\frac{65-50}{50} = \frac{15}{50} = 0.3$
$0.3 = 30\%$

4) New value after a percent change (multiplier method)

Use a multiplier to avoid extra steps.

• Increase by $r\%$: multiply by $1 + \frac{r}{100}$
• Decrease by $r\%$: multiply by $1 - \frac{r}{100}$

Example: Decrease $80$ by $15\%$.
$80\times \left(1-0.15\right)=80\times 0.85=68$

Trap check: “Decrease by $15\%$” is NOT the same as “is $15\%$ of.”

B) Ratio and proportion setup (the “label everything” method)

1) Build the ratio with consistent units

1. Write the comparison as a fraction.
2. Reduce if helpful.
3. Keep units aligned (miles with miles, dollars with dollars).

Example: $12$ girls to $18$ boys.
$\frac{12}{18} = \frac{2}{3}$

2) Solve a proportion

1. Write two equivalent ratios.
2. Cross-multiply.
3. Solve the resulting linear equation.

Example: $\frac{3}{5} = \frac{x}{20}$.
$3 \times 20 = 5x$
$60 = 5x$
$x=12$

Quick check: The answer should make the two ratios close in size.

C) “Part–part–whole” ratio word problems (tables help)

When you’re told something like $a:b$ and a total, use a parts table.

Steps

1. Add ratio parts: $a+b$.
2. Find value of 1 part: $\frac{\text{total}}{a+b}$.
3. Multiply by $a$ and $b$.

Example: Ratio of cats to dogs is $2:5$. Total pets $=49$. How many dogs?
$2+5=7$
$\text{one part} = \frac{49}{7}=7$
$\text{dogs} = 5\times 7 = 35$

D) Unit rate and “for every”

If a relationship is proportional, the ratio $\frac{y}{x}$ is constant.

Example: $\$18$ for $6$ tickets. Cost per ticket?
$\frac{18}{6}=3$
So $\$3$ per ticket.

E) Reverse percent (finding the original)

If you know the final amount after a percent change, divide by the multiplier.

Example: After a $20\%$ discount, the price is $32$. Original price?
$32 = \text{original}\times 0.8$
$\text{original} = \frac{32}{0.8} = 40$

3) Key Formulas, Rules & Facts

Percent essentials

Ratio and proportion essentials

Scaling facts that often hide inside ratio/proportion questions

• If lengths scale by $k$, then perimeters scale by $k$.
• If lengths scale by $k$, then areas scale by $k^2$.

Example statement (know the pattern):
$\text{area scale factor} = k^2$

Fraction–decimal–percent “anchors” (know instantly)

4) Examples & Applications

Example 1: Multi-step percent (discount then tax)

A jacket costs $80$. It’s discounted $25\%$, then sales tax of $8\%$ is applied to the discounted price. Final cost?

Setup using multipliers:
$80\times 0.75 = 60$
$60\times 1.08 = 64.8$
Final cost:
$64.8$

Key insight: Percent changes stack multiplicatively, not additively.

Example 2: Ratio split with a “difference” clue

The ratio of Mia’s to Noah’s points is $5:3$. Mia has $24$ more points than Noah. How many does Noah have?

Let points be $5x$ and $3x$.
Difference:
$5x-3x=2x=24$
$x=12$
Noah:
$3x=36$

Key insight: Ratio tells you relative sizes, and “more than” gives an equation.

Example 3: Proportional relationship from a table (constant of proportionality)

If $4$ notebooks cost $10$, how much do $14$ notebooks cost (assuming proportional)?

Unit rate:
$\frac{10}{4}=2.5$
Cost:
$14\times 2.5=35$

Key insight: Find the constant rate once, then scale.

Example 4: Similar figures hidden as a proportion

A rectangle has side lengths $6$ and $9$. A similar rectangle has the shorter side $10$. Find the longer side.

Scale factor:
$k=\frac{10}{6}=\frac{5}{3}$
Longer side:
$9\times \frac{5}{3}=15$

Key insight: Corresponding sides must match (short-to-short, long-to-long).

5) Common Mistakes & Traps

1. Mixing up “percent of” vs. “percent change”

   • What goes wrong: You use $\frac{\text{new}-\text{old}}{\text{new}}$ or multiply by the wrong base.
   • Why it’s wrong: Percent change is always relative to the original.
   • Fix: Say out loud: “Change divided by original.”

2. Using the wrong “whole” in part–whole questions

   • What goes wrong: You compute $\frac{part}{part}$ or use the wrong total.
   • Why it’s wrong: Percent is based on a specific reference.
   • Fix: Circle the phrase after “of” or identify what “out of” means.

3. Swapping ratio order

   • What goes wrong: A problem says boys:girls $=2:3$ and you write $\frac{3}{2}$.
   • Why it’s wrong: Ratios are ordered comparisons.
   • Fix: Keep labels: $\frac{\text{boys}}{\text{girls}}=\frac{2}{3}$.

4. Adding percents instead of multiplying multipliers

   • What goes wrong: You treat discount $25\%$ and tax $8\%$ as net $-17\%$.
   • Why it’s wrong: Each percent applies to a different base.
   • Fix: Use multipliers sequentially.

5. Cross-multiplying with mismatched quantities

   • What goes wrong: You set $\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}$ by accident.
   • Why it’s wrong: Units must correspond.
   • Fix: Align units in the same position in both ratios.

6. Forgetting that “percent points” are not the same as percent change

   • What goes wrong: You treat an increase from $20\%$ to $30\%$ as a $10\%$ increase.
   • Why it’s wrong: It’s a $10$ percentage-point increase, but percent change is relative.
   • Fix: Compute percent change:
     $\frac{30-20}{20}=0.5$
     $0.5=50\%$

7. Not reducing a ratio before splitting a total (or reducing incorrectly)

   • What goes wrong: You split by $2:6$ as if it’s different from $1:3$.
   • Why it’s wrong: Equivalent ratios represent the same split.
   • Fix: Reduce first or use the part-sum method consistently.

8. Rounding too early

   • What goes wrong: You round a unit rate like $\frac{10}{3}$ to $3.3$ and drift.
   • Why it’s wrong: Early rounding changes later results.
   • Fix: Keep fractions or more decimals until the end.

6) Memory Aids & Quick Tricks

Fast percent mental math (high-yield)

• $10\%$ of $N$ is $\frac{N}{10}$.
• $5\%$ of $N$ is half of $10\%$.
• $1\%$ of $N$ is $\frac{N}{100}$.
• Combine: $17\% = 10\%+5\%+2\%$ (and $2\%$ is twice $1\%$).

Example: $17\%$ of $240$.
$10\%: 24$
$5\%: 12$
$2\%: 4.8$
Total:
$24+12+4.8=40.8$

7) Quick Review Checklist

• You can translate percent as a decimal and use
  $\text{part} = \text{percent}\times \text{whole}$
• You always compute percent change with
  $\frac{\text{new}-\text{old}}{\text{old}}$
• You use multipliers for “after increase/decrease” problems:
  $\text{new} = \text{old}\times \left(1\pm \frac{r}{100}\right)$
• You keep ratio order and labels consistent (e.g., boys:girls).
• You solve ratio splits by adding parts, finding 1 part, then scaling.
• You set up proportions with matching units and cross-multiply cleanly.
• You sanity-check: Does your answer fit the context (bigger/smaller, reasonable size)?

You’ve got this: translate carefully, label your units, and let the algebra do the work.