Exam Sprint 1: Percent of, what percent, and the whole
Goal: solve the three basic percentage questions in under 30 seconds each, and know why each step works.
How this lesson works: you answer before you read. Then every worked example goes slowly: a story, what we know and need, 1% first, then 100%, a bar picture, and only then the shortcut.
Start from the basics
Before any sprint, make sure these four ideas are solid. Every question in this lesson uses them.
1. What is a percent? "Per cent" means per hundred. 35% means 35 out of every 100. Think of ₹100 as 100 coins of ₹1. If you spend 35 of the coins, you spent 35 out of 100, which is 35% of your money. On a 100-square grid each square is 1%, so 35% is 35 shaded squares:
███████████████████████████████████·································································
Text version: 35 of the 100 squares are shaded (35%); 65 are not (65%).
2. What does "of" mean? "Of" tells you which thing the percent is taken from. "35% of ₹200" means: cut ₹200 into 100 equal pieces (each piece is ₹200 ÷ 100 = ₹2) and take 35 pieces: 35 × ₹2 = ₹70.
3. Whole and part. The number after "of" is the whole: all of it, 100%. A part is a piece of the whole. The rest is what is left: whole − part, which is (100 − percent)%. In the example: whole ₹200, part ₹70 (35%), rest ₹130 (65%).
4. The one tool you will use in every question: find 1% first. 1% of anything = that thing ÷ 100. From 1% you can build any percent (multiply by the percent) or get back to the whole (multiply by 100). Try it now: 1% of ₹300 = 300 ÷ 100 = ₹3, so 35% of ₹300 = 35 × 3 = ₹105.
Words to know
- Percent (%)
- out of every 100 35% = 35 out of 100
- Whole
- the number after 'of' = 100% ₹200 in '35% of ₹200'
- Part
- a piece of the whole ₹70 = 35% of ₹200
- Rest
- whole − part ₹130 = 65% of ₹200
- 1%
- whole ÷ 100 1% of ₹200 = ₹2
1. Warm-up from memory
Warm-up: which fraction equals 25%?
Show answer
Correct answer: C
25% means 25 out of 100 = 1/4.
Option A: 1/25 is 4%.
Option B: 2/5 is 40%.
Option D: 1/5 is 20%.
Hint: Divide 25 and 100 by 25.
2. Predict first
Predict before reading: a coaching class has 80 seats and 20% are reserved for girls. How many seats are reserved?
Show answer
Correct answer: A
We know: 100% of the seats = 80. We need: 20% of the seats. Step 1: 1% of 80 = 80 ÷ 100 = 0.8. Step 2: 20% = 20 × 0.8 = 16 seats. Check: 16 reserved + 64 open = 80 ✓.
Option B: Percent read as units: 20 is the percent, not the number of seats.
Option C: Answered the remainder: 64 seats are the other 80% of the 80 seats.
Option D: Divided by the percent: 80 ÷ 20 = 4 divides by the percent.
Hint: Find 10% first.
3. Understand it with a story: finding the whole
A percent always means a share of something. 35% of your money means 35 parts out of every 100 parts of your money. So before any calculation, ask: 35% of what?
Question
21 is 35% of which number?
Solution
- What is the question asking?
- ₹21 is only a part. It is 35% of a bigger amount.
- We must find that bigger amount: the whole, which is 100%.
- Make it a story
- You spend 35% of your money on a book, and the book costs ₹21. How much money did you have at the start?
- We know / We need
- We know: 35% of your money = ₹21
- We need: 100% of your money = ?
- Step 1: find 1%
- 35% of your money = ₹21
- 35% means 35 equal pieces of 1%. Share ₹21 equally among 35 pieces.
- 1% of your money = 21 ÷ 35 = ₹0.6
- Step 2: find 100%
- 1% of your money = ₹0.6
- 100% is 100 such pieces: 100% of your money = 0.6 × 100 = ₹60
- See it on a bar
███████░░░░░░░░░░░░░each block = 5% of your money- █ 35% = ₹21 (spent on the book)
- ░ 65% = ₹39 (left with you)
- Whole bar = 100% = ₹21 + ₹39 = ₹60
- Shortcut (only after you understand)
- Whole = Part × 100 ÷ Percent = 21 × 100 ÷ 35 = 60
- Faster way (optional): 35% = 7 × 5%. So 5% of your money = 21 ÷ 7 = ₹3, and 100% = 20 × 5% = 20 × 3 = ₹60.
- Quick check
- 35% of ₹60 = 60 × 35 ÷ 100 = ₹21 ✓
- Part + rest = 21 + 39 = 60 ✓
Answer: ₹60, so the number is 60
Watch out: Most important concept: ₹21 is only a PART of your money. Work backward from the part to the whole (100%). Do not calculate 35% of 21: that gives 7.35, which is even smaller than the part.
Test your understanding: 15 is 25% of which number?
Show answer
Correct answer: B
We know: 25% of the number = 15. We need: 100% of the number. Step 1: 1% = 15 ÷ 25 = 0.6. Step 2: 100% = 0.6 × 100 = 60. Check: 25% of 60 = 15 ✓. 15 is only a part; work backward to the whole.
Option A: Took percent of the part: 3.75 is 25% of 15. But 15 is the part, not the whole: the answer must be bigger than 15.
Option C: Added the percent: 40 = 15 + 25 adds a number of units to a percent. They are different kinds of numbers.
Option D: Wrong number of quarters: 75 = 15 × 5. 25% is one quarter, so the whole is 4 × 15, not 5 × 15.
Hint: Find 1% of the number first: share 15 among 25 pieces.
4. The same idea: what percent?
Question
You scored 42 marks out of 60. What percent is that?
Solution
- What is the question asking?
- 60 marks is the whole (100%). 42 marks is a part of it.
- We must say how big the part is, counted in hundredths of 60 marks.
- Make it a story
- A class test has 60 marks in total. You scored 42. Your report card asks for a percentage.
- We know / We need
- We know: 100% of the total marks = 60 marks, and the part = 42 marks
- We need: 42 marks = ?% of the total marks
- Step 1: find 1% of the whole
- 1% of the total marks = 60 ÷ 100 = 0.6 marks
- Step 2: count how many 1% pieces fit in the part
- 42 ÷ 0.6 = 70 pieces of 1%
- So 42 marks = 70% of the total marks
- See it on a bar
██████████████░░░░░░each block = 5% of the total marks = 3 marks- █ 70% = 42 marks (your marks)
- ░ 30% = 18 marks (marks you lost)
- Whole bar = 100% = 60 marks
- Shortcut (only after you understand)
- Percent = Part ÷ Whole × 100 = 42 ÷ 60 × 100 = 70%, so 42 is 70% of 60
- Faster way (optional): 42/60 = 7/10 = 70/100 = 70%.
- Quick check
- 70% of 60 marks = 60 × 70 ÷ 100 = 42 marks ✓
Answer: 70%
Watch out: Most important concept: Divide by the WHOLE (60), not by the part. The part 42 is smaller than the whole, so the answer must be below 100%.
5. Now the shortcuts (you already know why they work)
Every basic percentage question links three numbers: the whole (100%), the percent, and the part. The question hides one of them. Each formula below is just the 1% → 100% idea written in one line:
The three questions
- Find the part
- percent × whole ÷ 100 20% of 80 = 16
- Find the percent
- part ÷ whole × 100 16 out of 80 = 20%
- Find the whole
- part × 100 ÷ percent 16 is 20% of 80
The whole is the number after of (or out of). If the whole is missing, the question says 'of which number'.
6. Now you finish it (half-solved)
Question
48 is 40% of the total marks. What are the total marks?
Solution
- What is the question asking?
- 48 marks is only a part. It is 40% of a bigger number.
- We must find that bigger number: the whole, which is 100%.
- Make it a story
- To pass an exam you need 40% of the total marks. The pass mark is 48.
- We know / We need
- We know: 40% of the total marks = 48 marks
- We need: 100% of the total marks = ?
- Step 1: find 1%
- 40% of the total marks = 48 marks
- 40% means 40 equal pieces of 1%. Share 48 marks equally among 40 pieces.
- 1% of the total marks = 48 ÷ 40 = 1.2 marks
- Step 2 is yours: find 100%
- You know 1% of the total marks = 1.2 marks. 100% is 100 such pieces.
- Work it out on paper first, then read the answer below.
- See it on a bar
████████░░░░░░░░░░░░each block = 5% of the total marks- █ 40% = 48 marks (needed to pass)
- ░ 60% = ? (the rest)
- Whole bar = 100% = ? (your answer)
- Shortcut (only after you understand)
- Whole = Part × 100 ÷ Percent = 48 × 100 ÷ 40
Answer: 120 marks. Step 2: 1% = 1.2 marks, so 100% = 1.2 × 100 = 120 marks. Bar: 40% = 48 marks + 60% = 72 marks = 120 marks. Check: 40% of 120 marks = 48 marks ✓
Watch out: Most important concept: 48 marks is only a PART of the total marks. Work backward from the part to the whole (100%). Do not calculate 40% of 48: that gives 19.2, which is even smaller than the part.
7. Your turn, no help
Your turn: 27 is what percent of 90?
Show answer
Correct answer: B
We know: 100% = 90 and the part = 27. Step 1: 1% of 90 = 0.9. Step 2: 27 ÷ 0.9 = 30 pieces of 1% → 30%. Shortcut: 27 ÷ 90 × 100 = 30%. Check: 30% of 90 = 27 ✓.
Option A: Guessed a nearby fraction: 33⅓% would need 30 out of 90; 27 is less than one third of 90.
Option C: Difference read as percent: 63 is 90 − 27, a difference, not a percent.
Option D: Wrong question: 24.3 is 27% of 90, a different question.
Hint: Simplify 27/90 by 9.
In 'what percent of', divide by the whole: the number after of. Students who divide the other way get answers above 100%. If the part is smaller than the whole, the answer must be below 100%.
10% of a number = move its decimal one place left. 5% = half of that 10%. 1% = move the decimal two places. So 35% of 60 = 3 × (10% of 60) + (5% of 60) = 18 + 3 = 21. Also x% of y = y% of x: 16% of 25 = 25% of 16 = 4.
8. Mixed check
Mixed check: which question asks for the WHOLE?
Show answer
Correct answer: D
In '24 is 8% of what number?', 24 is only a part and the number after 'of' is missing: that missing number is the whole. Step 1: 1% = 24 ÷ 8 = 3. Step 2: 100% = 300.
Option A: This asks for a part.
Option B: This asks for the percent.
Option C: This asks for a part.
Hint: Which word comes right after 'of'?
Write one line for every miss: question → trap name → the fix in five words. Trap names to use: wrong base, answered the remainder, added the percents, points vs percent. Return to this lesson: tomorrow (redo the mixed check), day 3 (practice test cold), day 7 (only your error-log questions). Today's skill: part, percent and whole.
Five exam-style questions are attached below this lesson. Every option has a reason: read why each wrong option is wrong, not only the right one.
Sign in to keep your progress. Sign in