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Section 8.1

Percent

When we multiply a number by 10, 100, 1000, …, we move the decimal point

once, twice, three,… times to the right respectively.

When we divide a number by 10, 100, 1000, …, we move the decimal point once,

twice, three,… times to the left respectively.

Changing a Percent to a Decimal. To change a percent to a decimal, move the

decimal point twice to the left, then drop the percent symbol.

Example 1. Change each percent to a decimal.

54%,

Solution.

60%,

5500%, 6.75%,

0.00123%

54% = 54. % = 0.54

60% = 60. % = 0.6

5500% = 5500. % = 55.00 = 55

6.75% = 0.0675

0.00123% = 0.0000123

Changing a Decimal to a Percent. To change a decimal to a percent, move the

decimal point twice to the right, then add the percent symbol.

Example 2. Change each decimal to a percent.

0.62, 0.2, 3, 0.045, 0.000012

Solution.

0.62 = 62. % = 62%

0.2 = 20%

3 = 3. = 300%

0.045 = 4.5%

1

0.000012 = 0.0012%.

Changing a Fraction to a Percent. To change a fraction to a percent, first, change

the fraction to a decimal, then the decimal to a percent.

Example3. Change each fraction to a percent.

4 3

7

3

1

,

,

,

,

5 8 16 125 6

Solution.

4

5

= 0.8 = 80%

3

= 0.375 = 37.5%

8

7

= 0.4375 = 43.75%

16

3

125

1

6

= 0.024 = 2.4%

= 0.166666 … . . = .16̅ = 16.6666 … % = 16. 6̅%

= 17%

(to nearest whole percent)

= 16.7%

(to the nearest tenth of a percent)

= 16.67%

(to the nearest hundredth of a percent)

= 16.667%

(to the nearest thousandth of a percent).

……………………………………………………………………………………………………………………………...

Applications. Many applications involving percent are based on the following

formula:

𝐴 𝑖𝑠 𝑃 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝐵

Or,

𝐴 = 𝑃 ∙ 𝐵,

2

where 𝑷 is the percent changed to a percent. Notice that “is” means = , and

“𝑜𝑓” implies multiplication.

Example 4.

𝑊ℎ𝑎𝑡 𝑖𝑠 4.25% 0𝑓 $264?

Solution. 𝐴 = .0425 ∗ 264 = $11.22

Example 5.

Solution.

𝟏

12 % 𝑜𝑓 𝑤ℎ𝑎t amount of money is $2.50?

𝟐

0.125𝐵 = 2.5, and this implies 𝐵 = $20.00.

Example 6. $2.76 𝑖𝑠 𝑤ℎ𝑎𝑡 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 $45.99?

Solution.

2.76 = 𝑃45.99, and this implies 𝑃 =

2.76

45.99

= 0.060013046 = 6%, to

the nearest whole percent.

……………………………………………………………………………………………………………………………

Sales Tax. We use the following formulas to solve problems involving the sales

tax:

𝑺𝒂𝒍𝒆𝒔 𝒕𝒂𝒙 𝒂𝒎𝒐𝒖𝒕 = 𝑻𝒂𝒙 𝒓𝒂𝒕𝒆(𝒄𝒉𝒂𝒏𝒈𝒆𝒅 𝒕𝒐 𝒂 𝒅𝒆𝒄𝒊𝒏𝒎𝒂𝒍) ∗ 𝑰𝒕𝒆𝒎𝒔′ 𝒔𝒄𝒐𝒔𝒕.

𝑻𝒐𝒕𝒂𝒍 𝒄𝒐𝒔𝒕 = 𝑰𝒕𝒆𝒎′ 𝒔𝒄𝒐𝒔𝒕 + 𝑺𝒂𝒍𝒆𝒔 𝒕𝒂𝒙 𝒂𝒎𝒐𝒖𝒏𝒕.

Example 7. Suppose that sales tax rate is 6.75%. If an item is priced at $29.99,

find the amount sales tax and the total cost.

Solution.

𝑆𝑎𝑙𝑒𝑠 𝑡𝑎𝑥 𝑎𝑚𝑜𝑢𝑡 = 𝑇𝑎𝑥 𝑟𝑎𝑡𝑒(𝑐ℎ𝑎𝑛𝑔𝑒𝑑 𝑡𝑜 𝑎 𝑑𝑒𝑐𝑖𝑛𝑚𝑎𝑙) ∗ 𝐼𝑡𝑒𝑚𝑠 ′ 𝑠𝑐𝑜𝑠𝑡 =

= 0.0675 ∗ 29.99 = $2.02, (to the nearest cent).

𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 = 𝐼𝑡𝑒𝑚′ 𝑠𝑐𝑜𝑠𝑡 + 𝑆𝑎𝑙𝑒𝑠 𝑡𝑎𝑥 𝑎𝑚𝑜𝑢𝑛𝑡 = $29.99 + $2.02 = $32.01.

Example 8. (Challenging Problem??!) The total cost to purchase an item is

$702.12. If the sales tax is 6.75%, find the original price of the item.

Answer: $657.72. Did you get this answer?

3

Discounts. We use the following formulas to solve the problems involving

discounts:

𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕 𝒂𝒎𝒐𝒖𝒏𝒕

= 𝒅𝒊𝒔𝒄𝒐𝒖𝒏𝒕 𝒓𝒂𝒕𝒆(𝒄𝒉𝒂𝒏𝒈𝒆𝒔 𝒕𝒐 𝒂 𝒅𝒆𝒄𝒊𝒎𝒂𝒍) ∗ 𝒐𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝒑𝒓𝒊𝒄𝒆

𝑺𝒂𝒍𝒆𝒔 𝒑𝒓𝒊𝒄𝒆 = 𝑶𝒓𝒈𝒊𝒏𝒂𝒍 𝒑𝒓𝒊𝒄𝒆 − 𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕 𝒂𝒎𝒐𝒖𝒏𝒕.

Example 9. A dictionary that normally sells for $16.50 is on sale at 40% off.

a. What is the amount of discount? Answer:$6.60

b. What is the dictionary’s sale price? Answer: $9.90

Note. To find the sale price, you can also do the following:

𝑆𝑎𝑙𝑒 𝑝𝑟𝑖𝑐𝑒 = 0.6 ∗ 16.50 = $9.90, (why?).

…………………………………………………………………………………………………………………….

The Percent of Decrease and Percent of Increase. To find the percent of

decrease we use the following formulas,

𝑷𝒓𝒆𝒄𝒆𝒏𝒕 𝒐𝒇 𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒆 =

𝒂𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒆

,

𝒐𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓

𝑷𝒓𝒆𝒄𝒆𝒏𝒕 𝒐𝒇 𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒆 =

𝒂𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒆

.

𝒐𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓

Note. In finding the percent of decrease, the original number is the larger

number, and in finding the percent of increase, the original number is the smaller

number.

Example 10. A FAX machine regularly sells for $380. The sale price is $266. Find

the percent decrease (the discount rate).

Solution. 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒 = 380 − 266 = $114,

4

𝑃𝑟𝑒𝑐𝑒𝑛𝑡 𝑜𝑓 𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑒 =

𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒 114

=

= 0.3 = 30%.

𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟

380

Example 11. In 2013, a popular park was visited by 123,000 visitors, while

in 2014, the same park was visited by 145,000, visitors. Find the percent of

increase in the number of visitors to the park from 2013 to 2014. Round off

the answer to the nearest tenth of a percent.

Solution. 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 = 145000 − 123000 = 22000,

𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒

22000

𝑃𝑟𝑒𝑐𝑒𝑛𝑡 𝑜𝑓 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 =

=

= 0.17886 …

𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟

123000

= 17.9%.

5

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