CCAT Mathematical Reasoning: Percentages

A percentage is a number expressed out of 100.

To convert a number to a percentage: simply multiply the number by 100.

1. 1 = 1 * 100 = 100%

2. 6 = 6 * 100 = 600%

3. 0.5 = 0.5 * 100 = 50%

4. 1 / 4 = (1 / 4) * 100 = 25%

To convert a percentage to a number: simply divide the number by 100.

1. 5% = 5 / 100 = 1 / 20 or 0.05

2. 100% = 100 / 100 = 1

3. 60% = 60 / 100 = 0.6

4. 0.8% = 0.8 / 100 = 0.008

You should have the following conversions at your fingertips:

5% = 1/20 = 0.05

10% = 1/10 = 0.1

20% = ⅕ = 0.2

25% = ¼ = 0.25

40% = ⅖ = 0.4

50% = ½ = 0.5

60% = ⅗ = 0.6

75% = ¾ = 0.75

80% = ⅘ = 0.8

100% = 1

Q1: What percentage is 7 of 35?

Answer: (7/35) * 100 = (⅕) * 100 = 20%

Q2: 15 is 60% of what number?

Answer:

Method 1:

Let the number be x. Then, 60% of x = 15,

or, (60/100)*x = 15,

Or, x = 15 * (100 / 60) = 100 / 4 = 25

Method 2:

60% = 60/100 = 0.6

Hence, the required number is 15 / 0.6 = 150 / 6 = 25

(This would be the fastest method)

Method 3:

60% = 60/100 = ⅗

Hence, the required number is 15 * (5/3) = 5 * 5 = 25

Q3: A home goods store sells 175 rugs in the first quarter of the fiscal year. If it sells 20% more rugs in the next quarter, how many rugs does it sell in the second quarter?

Answer:

Method 1:

Number of rugs sold in 1st quarter = 175

The increase in sales in the next quarter = 20%

Hence the increase (number) = 20% of 175

= (20/100) * 175

= (1 / 5) * 175

= 35

Hence the number of rugs sold in the next quarter = 175 + 35 = 210

Method 2:

Percentage of rugs sold in next quarter = 100 + 20 = 120% of rugs sold in the first quarter

Hence, number sold = 120% of 175

= (120 / 100) * 175

= (6 / 5) * 175

= 6 * 35

= 210

Q4: An automobile parts factory placed an order worth $10,000 for raw materials. They got a discount of 18% because they paid in advance. How much did they have to pay?

Answer:

Discount amount = 18% of 10,000

= (18/100) * 10000

= 18 * 100

= 1800

Hence, the price paid = listed price — discount

= 10,000–1800

= $8200

Q5: A store has a sale where all hats are sold at a discount of 35%. If the regular price of the hats is $40, how many hats could be bought at the sale price if a shopper spent $390?

Answer:

Discount amount on a hat = 35%

Hence, net price of a hat (after discount) = (100–35) = 65% of $40

= (65/100) * 40

= (65/10) * 4

= (65 * 4) / 10

Let’s not simplify this for now. The above is now the price (after discount) of one hat.

Hence, number of hats which can be bought for $390

= 390 / {(65 * 4) / 10}

= (390 * 10) / (65 * 4)

= (6 * 10) / 4

= 60 / 4

= 15

Q6: An elite army unit only accepts 5% of applicants. How many applicants should be processed if the unit plans to accept 25 people?

Answer:

We can do this using the Unitary Method. Note that 5% of applicants being accepted implies that if 100 candidates apply, finally 5 will be accepted.

If 5 are to be accepted, then number of applicants = 100

Hence, if 25 are to be accepted, number of applicants

= (100 / 5) * 25 = 500

Some problems will include a set of two different percentages to be applied one after another. For instance:

Q7: If a university accepts 40% of students that apply for entrance, and 75% of admitted applicants join, what percentage of total applicants actually join?

Answer:

Let’s say that 100 students apply for entrance.

Then, the number accepted = 40% = 40% of 100 = (40/100) * 100

Of this, 75% join.

Hence, the number who join = 75% of those accepted

= 75% of (40/100) * 100

= (75/100) * (40/100) * 100

= (75/100) * (40)

= (75 / 10) * 4

= 300 / 10

= 30

In this case, we have saved some time by simplifying only after reaching the bolded step.

Now, try the following problems on your own. Solutions are provided towards the end of this article.

Q-1: A hat store sells 900 hats in its first year. If the number of hats it sells in its second year is 50% higher, how many hats will it sell in its second year?

Q-2: A new car is purchased for $24,000. The value of the car depreciates by 20% in the first year, and by 15% in the second year. What is the value of the car after 2 years?

Q-3: 48 is what percentage of 192?

25%, 30%, 45%, 50%, 60%

Q-4: 24 is 12% of what number?

36, 288, 200, 20, 400

Q-5: A long distance race begins with 64 runners from Team A and 64 runners from Team B. 20 runners from Team A and 8 runners from Team B do not finish the race. What percentage of the runners who finish the race are from Team B?

52%, 56%, 60%, 64%, 68%

Solutions:

Q-1: A hat store sells 900 hats in its first year. If the number of hats it sells in its second year is 50% higher, how many hats will it sell in its second year?

Answer:

Hats sold in second year = (100 + 50) = 150% of hats sold in first year

= 150% of 900

= (150 / 100) * 900

= 150 * 9

= 1350

Alternate method:

Increase in number of hats sold = 50% of 900

= (1 / 2) * 900

= 450

Hence, hats sold in second year = 900 + 450

=1350

Q-2: A new car is purchased for $24,000. The value of the car depreciates by 20% in the first year, and by 15% in the second year. What is the value of the car after 2 years?

Answer:

At the end of the first year, the price of the car = 80% of its price at the start of the year

= 24,000 * (80 / 100)

= P1 (say)

At the end of the second year, the price of the car = 85% of its price at the start of the year

= P1 * (85 / 100)

= 24000 * (80 / 100) * (85 / 100)

= 24 * 8 * 85

= 24 * 680

= 16320

Q-3: 48 is what percentage of 192?

25%, 30%, 45%, 50%, 60%

Answer:

(48 / 192) * 100

= (1 / 4) * 100

= 25%

Q-4: 24 is 12% of what number?

36, 288, 200, 20, 400

Answer:

24 / 0.12

= 200

Q-5: A long distance race begins with 64 runners from Team A and 64 runners from Team B. 20 runners from Team A and 8 runners from Team B do not finish the race. What percentage of the runners who finish the race are from Team B?

52%, 56%, 60%, 64%, 68%

Answer:

Number of runners from Team A who finish = 64–20 = 44

Number from Team B who finish = 64–8 = 56

So, total number of finishers = 44 + 56 = 100

And, percentage of finishers from Team B = (56 / 100) * 100

= 56%

Note: Don’t get confused between the data given (non finishers) and the answer required (finishers).

Coming up next: Mathematical Reasoning Number Comparison: https://av-krishnan.medium.com/ccat-mathematical-reasoning-number-comparison-36c8f97cca1c

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