Proportional Reasoning with Percents — Challenge

A percent is a rate out of 100. Turn it into a multiplier and every percent problem becomes one multiplication.

What these problems cover

These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Percent — Parts per hundred. 37% = 37100 = 0.37.
Multiplier — The single number you multiply by. A 20% increase has multiplier 1.20.
Discount — An increase / a decrease applied to a starting price.
Net — A 3-D shape unfolded flat.
Base — The number being multiplied. In 53 the base is 5.
Percent increase — How much something grew, as a percentage of what it was.

Problem 1

A stadium seats 350 people and 8% of the seats are reserved for staff. How many seats are reserved?

Worked answer

8% means 8100, so take 8100 of 350.

8100· 350 = 8· 3.5 = 28 seats.

Problem 2

After a 15% increase, a bus fare is $69. What was the fare in dollars before the increase?

Worked answer

$60

The new fare is 115% of the old one, so 1.15x = 69 where x is the old fare.

x = 691.15 = 60 dollars.

Taking 15% off 69 would give 58.65, which is the common mistake: the increase was 15% of the smaller number, not of the larger.

Problem 3

A town's population fell by 10% one year and then rose by 10% the next, leaving 9,900 people. What was the population before the fall?

9,800
9,900
10,000
10,100
10,200

Worked answer

10,000

A fall of 10% multiplies by 0.90 and a rise of 10% multiplies by 1.10, so the two years together multiply by 0.90· 1.10 = 0.99.

If p is the starting population then 0.99p = 9900, so p = 10,000.

The two changes do not cancel: the rise is 10% of a smaller number than the fall was, which is why the town ends up 1% down.

Problem 4

What is 45% of 80?

Worked answer

Write the percent as a decimal: 45% = 0.45.

0.45· 80 = 36. As a check, 50% of 80 is 40, and 45% should be a little less.

Problem 5

A desk is priced at $250 and sales tax of 6% is added at the till. What is the total in dollars?

$253
$259
$262
$265
$271

Worked answer

$265

Adding 6% tax means paying 106% of the price, so multiply by 1.06 in one step.

1.06· 250 = 265 dollars. The tax alone is 15 dollars.

Problem 6

A tank holds 40 litres of a solution that is 25% acid. How many litres of pure acid must be added to make the solution 40% acid?

Worked answer

The tank starts with 0.25· 40 = 10 litres of acid in 40 litres of liquid.

Adding x litres of pure acid gives 10+x litres of acid in 40+x litres of liquid, and that ratio must be 0.40.

10+x40+x = 0.40 gives 10+x = 16+0.4x, so 0.6x = 6 and x = 10 litres.

Check: 20 litres of acid in 50 litres of liquid is indeed 40%.

Problem 7

A shirt sells for $36 after a 10% discount. What was its price in dollars before the discount?

$32.40
$38.00
$39.60
$40.00
$44.00

Worked answer

$40.00

Paying after a 10% discount means paying 90% of the original, so 0.90x = 36.

x = 360.90 = 40 dollars.

Adding 10% to 36 gives 39.60, which is not the answer, because the discount was taken from 40 and not from 36.

Problem 8

A bookseller sells a book for $96 and makes a profit of 20% on what it cost. What did the book cost in dollars?

$76.00
$76.80
$80.00
$84.00
$115.20

Worked answer

$80.00

Profit is measured against the cost, so the selling price is 120% of the cost: 1.2c = 96.

c = 961.2 = 80 dollars, and the profit is the remaining 16 dollars.

Taking 20% off 96 gives 76.80, which would be right only if the profit were 20% of the selling price.

Problem 9

12 is what percent of 48?

Worked answer

25%

The phrase asks for 1248 written as a percent.

1248 = 14 = 25100 = 25%.

Problem 10

Of 600 students, 45% chose music as an elective, and 30% of those students also chose art. How many students chose both music and art?

Worked answer

First find how many chose music: 0.45· 600 = 270 students.

The second percent is taken of that group, not of all 600: 0.30· 270 = 81 students.

Taking 30% of 600 would give 180, which counts students who never chose music.

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