Percentage change as a multiplier
Add a percentage on, or take it off, with a single multiplication.
Two statements, one meaning?
The population of a state was p in 1961. In 1991 it is 165% of p. Another way to say it: the population increased by 65% from 1961 to 1991.
Do the two statements mean the same thing?
What this lesson covers
The idea
Increasing a quantity by r% makes it (100 + r)% of the original, that is, multiplies it by 1 + r/100; decreasing it by r% leaves (100 – r)% of the original.
Two statements, one meaning?
The population of a state was p in 1961. In 1991 it is 165% of p. Another way to say it: the population increased by 65% from 1961 to 1991.
Do the two statements mean the same thing?
- Yes, they mean the same
- No, the first one is a bigger change
- No, the second one is a bigger change
Change an amount by a percentage
The top bar is the original amount, 100%. The bottom bar is the amount after the change. Change it by a percentage and read the multiplier. Make the changes in the goal.
One multiplier does it all
Increasing a quantity by r% makes it (100 + r)% of the original, that is, multiplies it by 1 + r/100; decreasing it by r% leaves (100 − r)% of the original.
Both statements about the population give the same result. If q is the population in 1991: Statement A: q = 165/100 × p = 1.65p Statement B: q = p + 0.65p = 1.65p The population in 1991 is 1.65 times the population in 1961.
A decrease by 20% leaves 100% − 20% = 80%, so we multiply by 0.8. A price of ₹500 becomes 0.8 × 500 = 400 rupees.
Notes
Increasing a quantity by r% makes it (100 + r)% of the original, that is, multiplies it by 1 + r/100; decreasing it by r% leaves (100 − r)% of the original.
Check yourself
A price of ₹200 increases by 30%. What is the new price? (in ₹)
Answer: 260
100% + 30% = 130% of the price, a multiplier of 1.3. 1.3 × 200 = 260, so ₹260.
The price of a shirt is ₹800. It is cut by 25%. What is the new price? (in ₹)
Answer: 600
100% − 25% = 75% is left, a multiplier of 0.75. 0.75 × 800 = 600, so ₹600.
A quantity decreases by 15%. By what number do we multiply it to get the new quantity?
A quantity is multiplied by 1.4. By what percentage has it increased?
Answer: 40 %
The multiplier 1.4 = 1 + 0.4, and 0.4 is 40%, so the quantity increased by 40%. It becomes 140% of the original.
- 0.85 — correct. Yes! 100% − 15% = 85% is left, so the multiplier is 0.85.
- 0.15. That is only the part that was taken away. What is left is 100% − 15% = 85%, so multiply by 0.85.
- 1.15. That is the multiplier for an increase of 15%. A decrease leaves 85%, so multiply by 0.85.