About Percentage
Percentage questions are comparison questions in disguise. A percentage is simply a fraction with 100 as its denominator, so every question in this topic is answered by deciding what the base is and then forming one fraction. Almost all the marks lost here come from choosing the wrong base, not from bad arithmetic.
What you need to understand
- A percentage is a number of parts out of every hundred, so x% of N is (x / 100) multiplied by N.
- The base is the quantity the percentage is measured against — the word "of" usually points at it.
- Percentage change is always measured against the ORIGINAL value, never the new one.
- Increasing a value by x% means multiplying it by (1 + x/100); decreasing means multiplying by (1 - x/100).
- To express one quantity as a percentage of another, divide the part by the whole and multiply by 100.
- A rise of x% followed by a fall of x% does not return to the starting value, because the two percentages are taken on different amounts.
Formulas to remember
How to work through these questions
- Identify the base first. Ask what the percentage is being taken of before writing any numbers down.
- Convert the percentage into a fraction or a multiplier immediately, so the rest of the work is ordinary multiplication.
- For change questions, put the original value in the denominator every time.
- For successive changes, multiply the factors rather than adding the percentages together.
- Check the direction: a percentage increase must produce a larger number and a decrease a smaller one.
Mistakes that cost marks
- Using the new value as the base in a percentage-change question instead of the original value.
- Adding successive percentages instead of multiplying the factors.
- Assuming that a rise of 20% followed by a fall of 20% brings the value back to where it started.
- Shifting the decimal point one place instead of two when converting between a percentage and a fraction.
- Reading "A is 25% more than B" as "B is 25% less than A" — the two statements have different bases.
Worked example
A number increases from 400 to 500. Find the percentage increase.
- The increase is 500 - 400 = 100.
- The base is the original value, which is 400.
- Percentage increase = (increase / original) x 100 = (100 / 400) x 100.
- 100 / 400 = 0.25, so the answer is 25%.
Answer: 25%
Practice questions with answers
A few Percentage questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
-
A
121.6
-
B
684
-
C
1216
-
D
12160
Answer: Option C — with explanation
64% means 64 parts out of every 100.
So 64% of 1900 = \(\frac{64}{100}\) x 1900 = 1216.
Common mistakes
- shifted the decimal point one place too far: 12160
- shifted the decimal point one place too little: 121.6
- took the remaining part instead of the part asked for: 684
Question 2
Express 408 as a percentage of 1200.
-
A
34
-
B
340
-
C
792
-
D
294.12
Answer: Option A — with explanation
Percentage = \(\frac{part}{whole}\) x 100.
= \(\frac{408}{1200}\) x 100 = 34%.
Common mistakes
- inverted the fraction: 294.12
- misplaced the decimal point in the division: 340
- subtracted the part from the whole instead of dividing: 792
Question 3
-
A
79.8
-
B
798
-
C
1102
-
D
7980
Answer: Option B — with explanation
42% means 42 parts out of every 100.
So 42% of 1900 = \(\frac{42}{100}\) x 1900 = 798.
Common mistakes
- shifted the decimal point one place too far: 7980
- shifted the decimal point one place too little: 79.8
- took the remaining part instead of the part asked for: 1102
Question 4
184 is what percent of 800?
-
A
434.78
-
B
23
-
C
616
-
D
230
Answer: Option B — with explanation
Percentage = \(\frac{part}{whole}\) x 100.
= \(\frac{184}{800}\) x 100 = 23%.
Common mistakes
- subtracted the part from the whole instead of dividing: 616
- misplaced the decimal point in the division: 230
- inverted the fraction: 434.78
Question 5
The price of an item rises from Rs. 700 to Rs. 896. Find the percentage increase.
Answer: Option B — with explanation
Percentage change is always measured against the ORIGINAL value.
Change = 896 - 700 = 196
Percentage = \(\frac{196}{700}\) x 100 = 28%.
Common mistakes
- gave the absolute change instead of the percentage: 196
- computed the complement of the percentage: 72
- measured the change against the new value instead of the original: 21.88
Frequently asked questions
Why is the percentage increase calculated on the original value?
Because the change happened to the original amount. Dividing by the new value would answer a different question, namely what fraction of the final figure the increase represents.
How do I handle two percentage changes in a row?
Multiply the two factors. A 10% rise followed by a 10% fall is 1.10 x 0.90 = 0.99, which is a net fall of 1%, not zero.
What is the quickest way to find x% of a number?
Convert the percentage to a decimal and multiply. For 35% of 240, work out 0.35 x 240 = 84. Dividing by 100 and then multiplying by 35 works just as well and is often easier by hand.
If A is 20% more than B, is B 20% less than A?
No. If B is 100 then A is 120, and B is 20 less than 120, which is 16.67% less. The base changed, so the percentage changed with it.
