On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I meet a percent question, the first thing I ask is: percent of what? I think about a day at the market. A seller says a bag of rice costs 20,000 francs, but today there is 10% off. I know 10% of 20,000 is 2,000, so I pay 18,000. The tricky part comes tomorrow. If the price goes up by 10%, it does not come back to 20,000, because now the 10% is taken from 18,000, which is only 1,800. The percent is always a part of one starting number, and that starting number can change. This is exactly what the SAT tests in Percentages. I find the starting amount first, and I check that every percent is taken from the right amount.
Percent questions look like they need clever tricks, but they only need one habit: name the starting amount. You can build that habit this week, and your scores will move.
What is inside this part
Finding a part when you know the whole and the percent (part = percent × whole).
20% of 350 · finding the whole from a part · percent of a group
Finding how much a value went up or down compared with its starting value, or finding the new value.
percent increase or decrease · discounts and markups · price after a change
Two or more percent changes in a row, where each new percent uses a new base.
increase then decrease · tax after discount · growth over several years
Questions where a percent of a substance stays the same amount while the total changes.
salt or acid solutions · adding water · mixing two percents
How to know a question is from this part
The question contains a percent sign or the words 'percent', 'discount', 'increase', 'decrease', 'tax' or 'of the total'.
- The % sign or the word 'percent' appears more than once.
- Words like discount, markup, tax, tip, interest, increase, decrease or 'percent more than'.
- A starting price or number is given, and you need a new price or a different percent.
- The words 'of those' or 'of these' that move from a big group to a smaller group.
- A solution that is 'x% salt' or 'x% acid' and something is added.
The lesson
1. What a percent really is
The word percent means 'out of 100'. So 25% means 25 out of every 100. It is the same as the fraction 25/100 and the decimal 0.25.
To change a percent into a decimal, move the decimal point two places to the left: 40% = 0.40, 5% = 0.05, 120% = 1.20. To change a decimal back, move two places to the right.
A percent is never a number on its own. It is always a percent OF something. That something is called the whole, or the base. Every percent question is really about finding the base.
2. The one formula behind everything
Almost every percent question is the same sentence: part = percent × whole. Write the percent as a decimal and multiply.
If you know any two of part, percent and whole, you can find the third. To find the percent: percent = part ÷ whole × 100. To find the whole: whole = part ÷ percent (as a decimal).
What is 35% of 80?
Show the steps
- Write 35% as 0.35.
- Multiply: 0.35 × 80 = 28.
28
3. Percent increase and percent decrease
When something goes up or down, the percent change tells us how big the change is compared with the STARTING value. percent change = (new − old) ÷ old × 100.
A positive answer is an increase. A negative answer is a decrease. Always divide by the old number, not the new one. This is the most common mistake.
A phone cost 50,000 francs last year and costs 40,000 francs now. What is the percent decrease?
Show the steps
- Change = 50,000 − 40,000 = 10,000.
- Divide by the old price: 10,000 ÷ 50,000 = 0.20.
- 0.20 is 20%.
20% decrease
4. The multiplier trick
A faster way is to use a multiplier. To increase a number by 15%, multiply it by 1.15 (that is 1 + 0.15). To decrease it by 15%, multiply by 0.85 (that is 1 − 0.15).
This also helps when you need to go backwards. If a price after a 20% discount is 48, then 0.80 × original = 48, so original = 48 ÷ 0.80 = 60. Going backwards means dividing by the multiplier.
A shirt costs 60 dollars after a 25% discount. What was the original price?
Show the steps
- A 25% discount means you pay 75% = 0.75 of the original.
- 0.75 × original = 60.
- original = 60 ÷ 0.75 = 80.
80 dollars
5. Percent changes in a row
If a number goes up 10% and then up 10% again, the total is NOT 20%. Multiply the multipliers: 1.10 × 1.10 = 1.21, so the total increase is 21%.
If it goes up 20% and then down 20%, you multiply 1.20 × 0.80 = 0.96. You end below where you started, down 4%. The second percent is taken from a different base.
A town has 5,000 people. It grows 20% and then shrinks 10%. How many people now?
Show the steps
- Multipliers: 1.20 and 0.90.
- 5,000 × 1.20 = 6,000.
- 6,000 × 0.90 = 5,400.
5,400 people
6. Percent of a percent, and 'percent of the group'
Sometimes the question gives a group, then a percent of that group, then a percent of the smaller group. Work one step at a time and write the new total after each step.
Example: 200 students, 60% play football, 25% of the footballers are girls. First 60% of 200 = 120 footballers. Then 25% of 120 = 30 girls. Never take 25% of 200 by mistake.
There are 400 learners. 30% walk to school. Of the learners who walk, 50% are in Senior 4. How many walk and are in Senior 4?
Show the steps
- Walkers: 0.30 × 400 = 120.
- Senior 4 walkers: 0.50 × 120 = 60.
60
7. Mixtures and 'percent solution' questions
In a mixture question, find the actual amount of the thing that matters (salt, sugar, acid). Amount = percent × total. The amount of salt does not change when you only add water.
Set up: salt amount ÷ new total = new percent. Solve for the unknown. This is a very hard type, but it is the same formula.
8. Common traps
Trap 1: using the new number as the base. The base for percent change is always the old number.
Trap 2: adding percents that should be multiplied (10% up then 10% up is 21%, not 20%).
Trap 3: reading 'percent more than' as 'percent of'. 150% of a number is 1.5 times it. 50% MORE than a number is also 1.5 times it. But 50% of a number is only half of it.
Trap 4: forgetting to answer the question that is asked. If it asks 'how many were NOT', subtract at the end. Underline what is asked before you calculate.
See it
- 1Underline what is asked (a part, a percent, or the whole).
- 2Name the starting amount (the base).
- 3Turn the percent into a decimal or a multiplier.
- 4Use part = percent × whole, or multiply by 1.xx or 0.xx.
- 5Check: does the answer make sense (bigger after an increase, smaller after a decrease)?
| Words in the question | Multiplier | Example |
|---|---|---|
| increased by 20% | × 1.20 | 50 becomes 60 |
| decreased by 20% | × 0.80 | 50 becomes 40 |
| 20% of a number | × 0.20 | 20% of 50 is 10 |
| 120% of a number | × 1.20 | 120% of 50 is 60 |
| after a 20% discount, price is P | original = P ÷ 0.80 | P = 40 gives 50 |
Each percent is taken from a different base, so you end at 19,800, not 20,000.
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