Finding the Percentage Increase or Decrease

The original value is his salary before the rise, $3500. The new value is $3710. In the drawing the original salary is the whole bar, 100%, and the increase is the extra piece at the end of the new bar.

$\text{Increase} = 3710 - 3500 = 210$, so the increase is $210.

Express the increase as a percentage of the original salary: $\frac{210}{3500} \times 100\% = 6\%$

Check by working forward. After a 6% increase the new salary is $100\% + 6\% = 106\%$ of the original salary, and $1.06 \times 3500 = 3710$. This is the new salary given in the question.

An increase of 6% means that the salary went up by $6 for every $100 of the original salary.

The original price is $1250 and the new price is $1050. In the drawing the new bar is shorter than the original bar, and the dashed piece is the decrease.

$\text{Decrease} = 1250 - 1050 = 200$, so the decrease is $200.

Express the decrease as a percentage of the original price: $\frac{200}{1250} \times 100\% = 16\%$

Check by working forward. After a 16% decrease the new price is $100\% - 16\% = 84\%$ of the original price, and $0.84 \times 1250 = 1050$.

The original value is last year's number of shows, 250. The new value is this year's number, 100.

$\text{Decrease} = 250 - 100 = 150$ shows

Express the decrease as a percentage of last year's number: $\frac{150}{250} \times 100\% = 60\%$

Check: after a 60% decrease, 40% of the shows are left, and $0.4 \times 250 = 100$. This is this year's number.

The original price is $4 and the new price is $5.

$\text{Increase} = 5 - 4 = 1$, so the increase is $1.

Express the increase as a percentage of the original price: $\frac{1}{4} \times 100\% = 25\%$

Ali divided by the new price: $\frac{1}{5} \times 100\% = 20\%$. That is the increase as a percentage of the new price, $5. A percentage increase is a percentage of the original price.

Working forward shows which answer is right. Increasing $4 by 20% gives 120% of $4, and $1.2 \times 4 = 4.8$, which is $4.80, not $5. Increasing $4 by 25% gives 125% of $4, and $1.25 \times 4 = 5$, which is $5.

The $54 is the new price, after the rise. The original price is not given, so find it first: $54 - 9 = 45$, so the original price was $45.

Express the increase as a percentage of the original price: $\frac{9}{45} \times 100\% = 20\%$

Check: 120% of $45 is $1.2 \times 45 = 54$, which is $54.

Dividing by the price given in the question, $\frac{9}{54} \times 100\% \approx 16.7\%$, would use the wrong base, because $54 is the new price.

$\text{Increase} = 1500 - 1300 = 200$ students

Express the increase as a percentage of the original enrolment: $\frac{200}{1300} \times 100\% = 15.384615\ldots\%$

Correct to 3 significant figures, $15.384615\ldots\% \approx 15.4\%$.

Do not round the fraction first. $\frac{200}{1300} = 0.153846\ldots$, and rounding it to 0.15 before multiplying by 100% gives 15%, which is not correct to 3 significant figures.

The original value is the number of visitors on the first Saturday, 1260.

$\text{Decrease} = 1260 - 1085 = 175$ visitors

Express the decrease as a percentage of the first Saturday's number: $\frac{175}{1260} \times 100\% = 13.888\ldots\%$

Correct to 3 significant figures, $13.888\ldots\% \approx 13.9\%$.

The original value is $120. In the drawing the new bar is more than twice as long as the original bar, so expect an answer of more than 100%.

$\text{Increase} = 300 - 120 = 180$, so the increase is $180.

Express the increase as a percentage of the original savings: $\frac{180}{120} \times 100\% = 150\%$

Check by working forward. After an increase of 150% the savings are $100\% + 150\% = 250\%$ of the original savings, and $2.5 \times 120 = 300$.

Do not mix up the two percentages. The new savings are 250% of the original savings, because $\frac{300}{120} \times 100\% = 250\%$. The increase is 150% of the original savings. The percentage increase is the second of these.

Write both prices in cents: $1.10 = 110 cents.

$\text{Increase} = 110 - 85 = 25$ cents

Express the increase as a percentage of the original price: $\frac{25}{85} \times 100\% = 29.411\ldots\%$

Correct to 3 significant figures, $29.411\ldots\% \approx 29.4\%$.

Write both times in seconds: 1 min 5 s = 60 s + 5 s = 65 s.

$\text{Decrease} = 65 - 52 = 13$ s

Express the decrease as a percentage of her time at the start of the year: $\frac{13}{65} \times 100\% = 20\%$

Do not write 1 min 5 s as 1.05 min. A minute has 60 seconds, not 100, so 1 min 5 s is $1\frac{5}{60} = 1\frac{1}{12}$ min.

Chicken rice: the increase is $4 - 3.5 = 0.5$, so $0.50. Its original price is $3.50.

Percentage increase in the price of chicken rice: $\frac{0.5}{3.5} \times 100\% = 14.285\ldots\% \approx 14.3\%$

Kopi: the increase is $1.4 - 1.2 = 0.2$, so $0.20. Its original price is $1.20.

Percentage increase in the price of kopi: $\frac{0.2}{1.2} \times 100\% = 16.666\ldots\% \approx 16.7\%$

The price of chicken rice rose by more money (50 cents against 20 cents), but the price of kopi rose by the greater percentage, because 20 cents is a larger part of $1.20 than 50 cents is of $3.50.