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Percent Problems: 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)

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Problem 1 of 10

Problem 1 The unused share

A fund uses $35\%$ of its money for supplies and $20\%$ for transport. The categories do not overlap. If the fund starts with $220$ dollars, how much remains for other uses?
1. Hint 1
  The used parts refer to the same original whole.
2. Hint 2
  Find the fraction of the whole left after both categories.
Answer

$99$ dollars.

Full solution

The two categories use $35\%+20\%=55\%$ , leaving $45\%$ .
The remaining amount is
$0.45(220)=99$
dollars.

Supplies use $77$ dollars and transport uses $44$ dollars; $77+44+99=220$ checks the total.

Answer

$99$ dollars.

Key idea

Percent parts of one whole that do not overlap can be added before the remaining part is found.
Problem 1: Mark complete
Problem 2 The paired portions

A bag holds only green and yellow counters. There are $12$ more yellow counters than green counters, and green counters are $35\%$ of the bag. Find the total number of counters.
1. Hint 1
  Express each color as a percent of the same whole.
2. Hint 2
  The difference is the yellow share minus the green share.
Answer

$40$ counters.

Full solution

Let $w$ be the total.
Since the bag holds only green and yellow counters, yellow is $100\%-35\%=65\%$ of the bag.
The green count is $0.35w$ and the yellow count is $0.65w$ .
Their difference gives
$0.65w-0.35w=12$ $0.30w=12$ $w=40$

There are $14$ green and $26$ yellow counters.
Their difference is $12$ , and $14/40=0.35$ , checking the stated share.

Answer

$40$ counters.

Key idea

A difference between two percent parts can determine their common whole.
Problem 2: Mark complete
Problem 3 The finished orders

A report lists $16$ completed orders and $9$ unfinished orders. Three of the unfinished orders are then completed. What percent of the original orders is now complete?
1. Hint 1
  The total number of orders does not change when an unfinished order is completed.
2. Hint 2
  Add to the completed count, then divide by the original total.
Answer

$76\%$ .

Full solution

There are $16+9=25$ original orders.
The updated completed count is $16+3=19$ , so
$r=\frac{19}{25}$ $r=0.76$
Thus $76\%$ are complete.

The remaining six orders form $24\%$ of the total, and the two shares add to $100\%$ .

Answer

$76\%$ .

Key idea

When items move between categories, update the part while keeping the shared whole fixed.
Problem 3: Mark complete
Problem 4 The service receipt

A repair receipt includes a $10\%$ increase on the original service charge and then a separate fixed delivery charge of $8$ dollars. The final total is $74$ dollars. Find the original service charge and the dollar amount of the increase.
1. Hint 1
  Only the increase, not the delivery charge, was computed as a percent of the service charge.
2. Hint 2
  Remove the separate fixed charge before undoing the percentage increase.
3. Hint 3
  The service charge after the increase is $1.10$ times its original amount.
Answer

Original service charge: $60$ dollars. Increase: $6$ dollars.

Full solution

Let $w$ be the original service charge in dollars.
$1.10w+8=74$ $1.10w=66$ $w=60$
The increase is $0.10(60)=6$ dollars.

The receipt checks as $60+6+8=74$ dollars.
The delivery charge is not part of the base used for the percentage increase.

Answer

Original service charge: $60$ dollars. Increase: $6$ dollars.

Key idea

Undo fixed charges and percent multipliers in the reverse order from which they were applied.
Problem 4: Mark complete
Problem 5 The two estimates

A supplier quotes $80$ dollars for a batch. Plan A increases the quote by $25\%$ and then reduces that new amount by $10\%$ . Plan B reduces the quote by $10\%$ and then adds a fixed $20$ dollars. Find each final amount and the difference between them.
1. Hint 1
  Each percent change multiplies the amount immediately before it.
2. Hint 2
  Keep the fixed addition in plan B separate from its percentage reduction.
Answer

Plan A: $90$ dollars. Plan B: $92$ dollars. Plan B is $2$ dollars more.

Full solution

Plan A uses the product of factors.
$80(1.25)(0.90)=90$
Plan B uses a factor followed by a fixed addition.
$80(0.90)+20=92$

The difference is $92-90=2$ dollars.
Adding the $20$ dollars before the reduction would give $(80+20)(0.90)=90$ , as in plan A; the $2$ dollar gap is the $10\%$ of $20$ that plan B never removes.

Answer

Plan A: $90$ dollars. Plan B: $92$ dollars. Plan B is $2$ dollars more.

Key idea

When a fixed addition and a percent change are combined, their order matters, because a percent change applied later also scales the added amount.
Problem 5: Mark complete
Problem 6 The missing reduction

A positive price is raised by $30\%$ and then reduced to $78\%$ of its original value. What percentage reduction was applied to the raised price? Check by starting with $100$ dollars.
1. Hint 1
  The final amount is compared with the original, but the reduction acts on the raised amount.
2. Hint 2
  Find the retained factor that multiplies $1.30$ to give $0.78$ .
Answer

$40\%$ ; the check runs from $100$ dollars to $130$ dollars to $78$ dollars.

Full solution

Let $f$ be the retained factor for the reduction.
The raise multiplies the price by $1.30$ and the reduction multiplies the raised price by $f$ , and together they must leave $0.78$ of the start, so
$1.30f=0.78$ $f=0.60$
Retaining $60\%$ means reducing by $40\%$ .

For the requested check, the raise takes $100$ dollars to $130$ dollars.
A $40\%$ reduction removes $52$ dollars and leaves $78$ dollars, the required $78\%$ of the start.

Answer

$40\%$ ; the check runs from $100$ dollars to $130$ dollars to $78$ dollars.

Key idea

A missing percent change is found by comparing the overall factor with the factors already known.
Problem 6: Mark complete
Problem 7 The interest record

An account holds a fixed principal that earns simple interest at a fixed annual rate, with no deposits or withdrawals. The account's balance is $456$ dollars at the end of year $2$ and $540$ dollars at the end of year $5$ . Find the principal and the annual rate.
1. Hint 1
  Simple interest adds the same amount each year while the principal stays fixed.
2. Hint 2
  The change in balance from the end of year $2$ to the end of year $5$ covers three years of interest.
3. Hint 3
  Take two years of interest away from the year $2$ balance to reach the principal.
Answer

Principal: $400$ dollars. Rate: $7\%$ per year.

Full solution

From the end of year $2$ to the end of year $5$ , three years of interest are added, and they total $540-456=84$ dollars.
The principal is fixed, so each year adds the same interest,
$\frac{84}{3}=28$
dollars.

By the end of year $2$ , two years of interest, $2(28)=56$ dollars, had been added to the principal.
So the principal is
$P=456-56$ $P=400$
dollars.

The annual rate is one year's interest divided by the principal.
$r=\frac{28}{400}$ $r=0.07$
which is $7\%$ per year.

As a check, $400(1+0.07\cdot 2)=456$ and $400(1+0.07\cdot 5)=540$ , the two recorded balances.

Answer

Principal: $400$ dollars. Rate: $7\%$ per year.

Key idea

An increase across a known interval reveals the constant annual addition in simple interest.
Problem 7: Mark complete
Problem 8 Dana's checkout claim

A first store takes $15\%$ off a positive price and then adds $8\%$ sales tax to the discounted price. A second store adds $8\%$ sales tax to the same price first and then takes $15\%$ off the taxed amount. Dana says the second store charges more because its discount is taken from a larger amount. Is she correct? Explain for any positive price.
1. Hint 1
  Each change multiplies the amount immediately before it.
2. Hint 2
  Let $w$ be the original price and write each store's final price as $w$ times two factors.
3. Hint 3
  Compare the two products of factors, not only the dollar sizes of the two discounts.
Answer

No; both stores charge $0.918$ times the original price, so their final prices are equal.

Full solution

Let $w>0$ be the original price.
Taking $15\%$ off multiplies by $0.85$ , and adding $8\%$ tax multiplies by $1.08$ .

The first store charges
$1.08(0.85w)=0.918w$
and the second store charges
$0.85(1.08w)=0.918w$
The two products agree because the order of multiplication does not change a product.

The second store's discount is indeed larger in dollars, $0.15(1.08w)=0.162w$ against $0.15w$ , but its tax is larger too, $0.08w$ against
$0.08(0.85w)=0.068w.$
Each extra amount is $0.012w$ , so they cancel, neither store charges more, and Dana is not correct.

Answer

No; both stores charge $0.918$ times the original price, so their final prices are equal.

Key idea

Successive percent changes give the same result in either order, because their factors multiply and a product does not depend on order.
Problem 8: Mark complete
Problem 9 Leila's savings claim

A store cuts a price by $30\%$ . Leila says the amount saved is then $30\%$ of the sale price as well. Is this true for a positive original price? State the saved amount as a fraction of the sale price.
1. Hint 1
  Write the savings and sale price as multiples of the same original price.
2. Hint 2
  Compare those two quantities by division.
Answer

No; the savings are $\frac37$ of the sale price.

Full solution

If the original price is $w>0$ , the savings are $0.30w$ and the sale price is $0.70w$ .
Their ratio is
$\frac{0.30w}{0.70w}=\frac37$
The denominator is nonzero because $w>0$ .

The fraction $3/7$ differs from $30/100$ , so the claimed $30\%$ comparison to the sale price is false.

Answer

No; the savings are $\frac37$ of the sale price.

Key idea

A percentage comparison changes when its reference whole changes.
Problem 9: Mark complete
Problem 10 The waiting period

Two accounts have the same positive principal and the same positive simple annual interest rate, with no deposits or withdrawals. Account B stays open twice as long as account A. Jules claims B earns twice the interest but does not have twice the final balance. Is this correct? Explain.
1. Hint 1
  The principal is unchanged in both accounts.
2. Hint 2
  Write each interest amount and then add the principal separately to each.
Answer

Yes.

Full solution

Let the principal be $P>0$ , annual rate $r>0$ , and time for A be $t>0$ .
Its interest is $Prt$ .
B earns
$Pr(2t)=2Prt$
which is twice the interest.

The balances are $P+Prt$ and $P+2Prt$ .
Twice A's balance is $2P+2Prt$ , which exceeds B's balance by the positive amount $P$ .
Therefore B does not have twice the balance, and both parts of the claim are true.

Answer

Yes.

Key idea

Simple interest scales with time while the principal portion of the balance stays fixed.
Problem 10: Mark complete

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