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Percentages
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Read the last sentence first
- Start:
- The last sentence, where it asks for the part, the percent, or the original.
- Stop:
- After you have found what is asked and which number is the base.
- Skip:
- Stories about the shop, school or town until you need a number from them.
- Why:
- The question line tells you whether to multiply (find a part) or divide (find the original), so you do not mix them up.
Used in questions 1, 2, 3, 6
2. Divide by the old number
- Start:
- The two numbers in a 'from ... to ...' sentence.
- Stop:
- When you have written (new − old) ÷ old.
- Skip:
- Any other numbers that are not the old or new value.
- Why:
- Percent change always uses the starting value as the base, so using the old number is always safe.
Used in questions 2, 3
3. Turn every percent into a multiplier
- Start:
- Each percent in the question, one at a time.
- Stop:
- When you have a chain like 1.20 × 0.75.
- Skip:
- Adding percents together. Never add percents that act in a row.
- Why:
- A chain of multipliers gives the exact result of successive changes without any guessing.
Used in questions 4, 5, 7
4. Going backwards means dividing
- Start:
- The words 'after' or 'was' near a discount or increase.
- Stop:
- When you have final price ÷ multiplier.
- Skip:
- Multiplying the final price by 1.25 or similar. It is a trap.
- Why:
- The final price is already a percent of the original, so you divide to undo it.
Used in questions 3, 5, 7
5. Write the group after each step
- Start:
- The biggest group in the story.
- Stop:
- After each 'of those' phrase, write the new group size.
- Skip:
- The temptation to take the second percent from the first total.
- Why:
- 'Of those' always means the smaller group, so write its size and use it as the new base.
Used in questions 4, 6
6. Hold the amount that does not change
- Start:
- The thing that stays the same in a mixture (salt, acid), found with percent × total.
- Stop:
- When you solve amount ÷ new total = new percent.
- Skip:
- Percent differences like 15% − 10%. They have no meaning here.
- Why:
- Water changes the total but not the amount of salt, so the salt equation is always true.
Used in question 8
7. Calculator check
- Start:
- Desmos or the Bluebook calculator after you set up the multiplier chain.
- Stop:
- When the display gives the number and you compare it with the options.
- Skip:
- Doing long decimals in your head.
- Why:
- Typing 119/1.19 or 54000/(1.2*0.75) is fast and removes arithmetic mistakes. Also test each option by plugging it back in.
Used in questions 3, 5, 7
8. Find the right whole before dividing
- Start:
- The words 'of the', 'among' or 'who' right before the group you are asked about.
- Stop:
- When you can name the total that goes at the bottom of the fraction.
- Skip:
- Totals in the table that belong to other groups.
- Why:
- Most wrong answers on table-percent questions use the grand total when the question names a smaller group, like 'of the students who walk'.
Used in questions 9, 10, 12
9. Percent of a percent: multiply, then add the pieces
- Start:
- The share of each group and the percent inside each group.
- Stop:
- When you have group share × inside percent for every row, then added them.
- Skip:
- Averaging the inside percents, because the groups have different sizes.
- Why:
- Each product is a real part of the whole country, so the sum is the true overall percent.
Used in questions 14, 16
10. Mixture: one equation on the pure amount
- Start:
- The table's volume and percent for each solution.
- Stop:
- When you have 'amount of acid in A + amount of acid in B = percent × total'.
- Skip:
- Averaging the percents 20% and 50%.
- Why:
- Acid is never lost when you mix, so the acid equation is always true, even when the percents are not simple.
Used in questions 15, 8