Core practice ← Back to lesson

Percent Increase and Decrease: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

0 of 10 completed

Progress saved in this browser.

Problem 1 of 10

Problem 1 Delivery route

A revised delivery route is $\frac{5}{6}$ as long as the original route. Describe the change in length as a percent increase or decrease, rounded to the nearest tenth of a percent.
1. Hint 1
  Compare the missing share with one whole original route.
2. Hint 2
  Convert that share of the original into a percent.
Answer

A decrease of approximately $16.7\%$ .

Full solution

The route lost one sixth of its original length.
$1-\frac{5}{6}=\frac{1}{6}$

Convert that original-length share to a percent.
$\frac{1}{6}\times100\%\approx16.7\%$
The route is shorter, so this is a decrease of approximately $16.7\%$ .

Answer

A decrease of approximately $16.7\%$ .

Key idea

Measure the share lost against the original whole when describing a percent decrease.
Problem 1: Mark complete
Problem 2 Machine report

A machine report says that the new output is $1.075$ times the old output, which was positive. What percent increase or decrease does this describe?
1. Hint 1
  A multiplier of one would leave the original amount unchanged.
2. Hint 2
  Find the part of the multiplier above or below one and express that part as a percent.
Answer

A $7.5\%$ increase.

Full solution

The multiplier keeps one full copy of the old output and adds
$1.075-1=0.075$
of it.

That added share is
$0.075=7.5\%$
of the old output, so the report describes a $7.5\%$ increase.

Answer

A $7.5\%$ increase.

Key idea

The amount by which a multiplier exceeds one gives the added share of the original.
Problem 2: Mark complete
Problem 3 Enlarged image

An image has been enlarged to $124\%$ of its original width and is now $62$ cm wide. What was its original width?
1. Hint 1
  The new width is more than one full copy of the original.
2. Hint 2
  Undo the multiplication that turns the original width into the new width.
Answer

$50$ cm.

Full solution

The new width is $1.24$ times the original.
Since this multiplier is not zero, divide to recover the original width.
$62\div1.24=50$

The original width was $50$ cm.
Checking forward, $1.24\times50=62$ , as required.

Answer

$50$ cm.

Key idea

Recover an original amount by dividing the new amount by its nonzero multiplier.
Problem 3: Mark complete
Problem 4 Repair shop

A shop pays 48 dollars for a used lamp and spends 12 dollars repairing it. The shop marks up its total cost by $25\%$ to set the selling price. What is the selling price?
1. Hint 1
  The markup is based on both parts of the shop's cost.
2. Hint 2
  Build a multiplier that keeps the full cost and adds one quarter of it.
Answer

75 dollars.

Full solution

The shop's total cost is
$48+12=60$
dollars.
A twenty-five percent markup makes the selling price $125\%$ of that cost.

The multiplier is $1.25$ , so the price is
$1.25\times60=75$
dollars.
The added 15 dollars is one quarter of the total cost, confirming the markup.

Answer

75 dollars.

Key idea

A markup adds its stated percent to the cost base named in the problem.
Problem 4: Mark complete
Problem 5 Catering invoice

A catering invoice lists food at 120 dollars and delivery at 20 dollars. Tax is $5\%$ of the food charge only. A tip is $10\%$ of the delivery charge only. What total is paid?
1. Hint 1
  Each added percent belongs to its own stated charge.
2. Hint 2
  Find the taxed food amount and the delivery amount with its tip, then combine them.
Answer

148 dollars.

Full solution

Tax increases the food charge by five percent, so that part of the payment is
$1.05\times120=126$
dollars.

The tip increases the delivery charge by ten percent, making that part
$1.10\times20=22$
dollars.
The total is
$126+22=148$
dollars.
The additions are 6 dollars of tax and 2 dollars of tip, matching the final total.

Answer

148 dollars.

Key idea

Apply each percent increase to the specific amount named as its base.
Problem 5: Mark complete
Problem 6 Two coupons

A membership normally costs 180 dollars. Coupon A takes $20\%$ off that price; Coupon B takes 32 dollars off. Only one coupon may be used, and there are no other charges. Which coupon gives the lower price, and by how much?
1. Hint 1
  Compare the resulting prices rather than the numbers printed on the coupons.
2. Hint 2
  The percent coupon leaves eighty percent of the listed price.
Answer

Coupon A gives a price 4 dollars lower.

Full solution

Coupon A leaves eighty percent of the listed price, a price of
$0.80\times180=144$
dollars.
Coupon B leaves a price of
$180-32=148$
dollars.

Coupon A therefore gives the lower price, with a difference of
$148-144=4$
dollars.

Answer

Coupon A gives a price 4 dollars lower.

Key idea

A percent discount and a fixed discount can be compared by calculating both prices from the same original.
Problem 6: Mark complete
Problem 7 Project award

A project's original award is increased by $20\%$ . A separate 10-dollar contribution is then added, bringing the total to 250 dollars. What was the original award?
1. Hint 1
  The 10 dollars was added after the percent increase, so it is not part of the increased award.
2. Hint 2
  Remove the 10 dollars first; what is left is one hundred twenty percent of the original award.
Answer

200 dollars.

Full solution

Remove the separate contribution first.
$250-10=240$
This is $120\%$ of the original award.

Divide by the nonzero increase multiplier.
$240\div1.20=200$
The original award was 200 dollars.
A twenty-percent increase adds 40 dollars, and the separate 10 dollars then brings it to 250 dollars.

Answer

200 dollars.

Key idea

Remove a fixed amount added after a percent change before dividing by the multiplier.
Problem 7: Mark complete
Problem 8 Permit charges

A permit has a positive listed price. The agency takes $25\%$ off the listed price for early filing and adds a $4\%$ processing charge, also calculated from the listed price. Imani says the permit costs $79\%$ of the listed price. Is she correct? Does your answer depend on what the listed price is? Justify your answer.

For one permit, the early-filing discount is 30 dollars. What does that permit cost?
1. Hint 1
  Identify the base each percent is taken from.
2. Hint 2
  Start from the whole listed price, remove one share of it and add the other.
3. Hint 3
  A known discount amount and its percent together recover the listed price.
Answer

Yes, she is correct, and the answer does not depend on the listed price; this permit costs 94.80 dollars.

Full solution

The early-filing discount and the processing charge are both calculated from the same listed price, so their percents of that price can be combined directly.

The permit keeps the full listed price, loses twenty-five percent of it and gains four percent of it.
$100\%-25\%+4\%=79\%$
Imani is correct; no particular price is needed because both percents share the same base.

The 30-dollar discount is one quarter of the listed price, so the listed price is
$30\div0.25=120$
dollars.
The permit keeps seventy-nine percent of it.
$0.79\times120=94.80$
The permit costs 94.80 dollars.

Answer

Yes, she is correct, and the answer does not depend on the listed price; this permit costs 94.80 dollars.

Key idea

Percents taken of the same base can be combined by adding and subtracting them.
Problem 8: Mark complete
Problem 9 Price restoration

A price rises from 55 dollars to 88 dollars. Rowan says that cutting the new price by the same percent as the rise will restore 55 dollars. Is Rowan right? What percent cut of the new price would restore the old price?
1. Hint 1
  The rise and the restoring cut begin from different prices.
2. Hint 2
  Compare the same dollar difference with fifty-five for the rise and with eighty-eight for the cut.
Answer

No; the cut that restores 55 dollars is $37.5\%$ , not $60\%$ .

Full solution

The rise is the new price minus the old price.
$88-55=33$
Measured against the original 55 dollars, that is
$\frac{33}{55}=60\%$
A sixty percent cut keeps forty percent of eighty-eight dollars, leaving
$0.40\times88=35.20$
dollars, not 55 dollars.

The required cut is the same 33 dollars measured against the new starting price of 88 dollars.
$\frac{33}{88}=37.5\%$
A $37.5\%$ cut keeps $62.5\%$ of eighty-eight dollars.
$0.625\times88=55$
This restores the old price.

Answer

No; the cut that restores 55 dollars is $37.5\%$ , not $60\%$ .

Key idea

The percent that reverses a change is measured against the changed amount, so it generally differs from the first percent.
Problem 9: Mark complete
Problem 10 Workshop attendance

Workshop attendance falls by $40\%$ to $48$ people. At the next session, attendance rises by $50\%$ of that reduced count. Jae says the final attendance is $10\%$ below the count before either change. Is Jae correct? Give the final attendance and justify your answer.
1. Hint 1
  The reduced count of 48 is a known percent of the first count.
2. Hint 2
  Recover that first count, then calculate the final count and compare the two, or combine the two multipliers into one and compare it with one.
Answer

Yes; the final attendance is $72$ people, a $10\%$ decrease overall.

Full solution

The fall leaves sixty percent of the first attendance count, so that first count is
$48\div0.60=80$
people.

The next session has fifty percent more people than the reduced count.
$1.50\times48=72$
This is eight fewer than the original eighty.
$\frac{80-72}{80}=0.10$
Jae is correct: the overall decrease is $10\%$ , and the final attendance is $72$ people.
The combined multiplier confirms this without the first count.
$0.60\times1.50=0.90$
Keeping ninety percent of the starting count is a $10\%$ decrease.

Answer

Yes; the final attendance is $72$ people, a $10\%$ decrease overall.

Key idea

When the second percent change is taken of the result of the first, their multipliers combine into one, the product of the two, so a $40\%$ fall then a $50\%$ rise leaves $0.90$ of the starting count.
Problem 10: Mark complete

← Back to lesson