How Many Out of a Hundred?

That's exactly what "per-cent" means -- PER HUNDRED. So a percentage is just a fraction with 100 in the denominator. If you get 18 out of 20 on a quiz, that's the same as 90%. Why? Because of equivalent fractions.

If a study finds that 380 out of 500 participants enjoy bike riding, their report will state that 76% of the respondents said yes to this question. In this case, we divide both 380 and 500 by 5, since we want the denominator to equal 100.

Changing Decimals to Percents

Changing a 2-place decimal to a percentage is simple because it already represents a fraction with a denominator of 100.

So, 0.76 = 76%.......and.......0.08 = 8%.

With more than 2 decimal places, we put a decimal in the percentage equivalence like this:

0.127 = 12.7%.......and.......0.0264 = 2.64%

Watch Out For numbers with just 1 decimal place. Lots of students make the mistake of writing 0.5 as 5% when it is really 50%, since 0.5 = 5/10 which is 50/100, the same as 0.50. When we have 0.05, it is 5% because it means 5/100.

Changing Whole and Mixed Numbers to Percent

Since a percentage is a fraction with a denominator of 100, and we know that 1 = 100/100, we can write 1 as 100%. And, we can write 7 as 700%. If you answered every question on a math test correctly, you will get 100% on that test. A winning bet on a horse that pays 2 to 1 will get you 200% or twice the amount you bet. A $10 bet on this horse would pay $20.

It should be obvious by now that we change any whole or decimal number to a percent by moving the decimal point 2 places to the right.

Note: if we include the decimal point, it must be followed by at least 1 digit.
So, 0.45 = 45% or 45.0%. We don't write 45. % -- with no digit after the decimal.

Examples: Change these numbers to percents:

a) 2.17 = 217%

b) 1.1234 = 112.34%

c) 0.0673 = 6.73%

d) 0.195 = 19.5%

Fractions to Percent

To change any fraction, no matter what its denominator, to a percent, we simply perform the division indicated by the fraction. Then we express the result as a percent, like this:

Example: Bart scored a goal in 31 of the 53 hockey games he played last season. His cousin Andrew scored a goal in 40 of his games, but he played 71 games. Which of the 2 had a better scoring record?

Solution: It's hard to tell if 31 out of 53 is bigger or smaller than 40 out of 71, so we change them both to percents and then it's easy to compare.

Bart's scoring record was better than Andrew's.

.

Finding Percent of a Number

The opinion poll results shown here were calculated from 23 750 responses received by e-mail. From these percents we can calculate that 22 249 people answered YES,
because 0.9368 × 23 750 = 22 249.

To find a given percent of a number, we multiply it by the decimal equivalent of the percent.

So, 53% of 927 = 0.53 × 927 = 491.31
6.3% of 17 = 0.063 × 17 = 1.071

Sometimes, we're given an amount and told it is a given percentage of a total that we have to find. Say we have to pay 5% interest on a loan and it amounts to $250. We need to find the amount of the loan (principle). Our question becomes 5% of ___ = $250. In such a case, we divide both sides of the equation by 5% or 0.05 to find the missing value.
We get 250 ÷ 0.05 = $5000.

Example: What was the amount of the loan if the interest rate is 7.6% and the interest owing on the loan after 1 year is $266?

Solution: 7.6% of ___ = $266. We divide through by 0.076. 266 ÷ 0.076 = 3500.
We borrowed $3500 at 7.6%.

Discounts and Sale Prices

The ad shown here announces a discount on ladies clothing. Gowns are selling at 25% to 50% off. This means that for a gown regularly priced at $325 with a 25% discount, we would only pay 75% of $325. We could find the sale price by first finding the discount and then subtracting that from $325, however, it's faster to find 75% of $325 since that's what we have to pay. Let's do it both ways and show we get the same result.

First, the long (wrong) way:

0.25 × 325 = $81.25
$325.00 - $81.25 = $243.75

Now the smart way:

0.75 × 325 = $243.75

So, if the question doesn't ask for the amount saved, just the sale price, we subtract the discount percent from 100% to find what part of the regular price we have to pay. Had there been a 37% discount on the gown we chose, we would find the sale price this way:

100% - 37% = 63%
0.63 × 325 = $204.75

Interest on Investments

$200 "Chief Joseph" U.S. Savings Bond

One way to safely invest money is to buy Savings Bonds. These are documents, issued by a government to raise money for public projects. They are guaranteed, so we can cash them in at any time -- however they pay very low interest. The bond seen here honoring Chief Joseph is worth $200 and pays 4.52% interest per annum (per year). It has a term of 5 years which means that it will only earn interest for 5 years from the date it is issued or printed.

Say we bought this bond when it was issued and have kept it for 5 years. When we cash it in, we will get the $200 face value plus the 5 years worth of interest. Since the bond pays 4.52% interest for 5 years, we will get an additional $45.20.

$200 × 0.0452 × 5 = $45.20

Investment Vocabulary

When we earn interest on an investment, we use special words to describe the money we invest, the interest we earn and the time that the money is invested. The money amount is called the Principle. The amount of interest we earn is expressed as a percent and we call it the rate. The time interval during which the investment is earning interest is called the term.

To calculate the simple interest on on an investment or loan, we use the formula:

I = P r t

I = interest, ..... P = principle, ..... r = rate,..... and ..... t = time or term

Example: How much interest will we earn in 3 years if we invest $7500 at 5.4% per annum?

Solution: I = P r t = $7500 × 0.054 × 3 = $1215.00. We will earn $1215.00 in 3 years.

Terrible Taxes

Unfortunately, taxes make up a large part of the money we spend on just about everything. The taxes we pay on the things and services we buy are called value added taxes. In Canada and Quebec, both the federal and provincial govenments collect these taxes "at point of sale" -- when you pay for the things or services. The 7% GST (Goods and Services Tax) is the federal tax, and the 8% PST is the provincial tax. When we pay for a meal or for items we've collected in a store, the cashier punches up the purchase total and then the cash computer adds on the GST and PST.

Note: With these taxes, we pay tax on tax. The first subtotal is the total amount for purchases plus the 7% GST. Then, the PST is calculated by taking 7% of this subtotal. So we pay provincial tax on federal tax.

Example: Find the total bill after taxes are added for purchases amounting to $27.50.

Solution: Here's what the checkout receipt will look like:

Here's what the computer did:
First, it added 7% of 27.50 to 27.50 to get $29.43.
Then it added 8% of 29.43 to 29.43 to get a total of $31.79.

If we don't need to show how much tax we're paying, there's a shorter way.
First we multiply 27.50 by 1.07 to get $29.43,
then we multiply this by 1.08 to get a total of $31.78.

When filling out most businesses forms we need to show the amount of tax paid, so this method is not used commercially.

Percent Increase or Decrease

Sometimes, we need to know by what percent a number increased or decreased. To find this percent, we compare the difference in the 2 numbers to the original number.

Example: The number of students enrolled at LearnStuff High School increased from 781 to 906 in the last school year. What percent increase was this?

Solution: First we find the number of new students, then we find what % it is of last year's enrollment.

The number of new students is 906 – 781 = 125,
so the percent increase is 125 ÷ 781 = 16.005% or 16%

Example: The number of year books sold in the school however dropped from 195 last year to 178 this year. What was the % decrease in the sale of yearbooks?

Solution: First we find the difference, then determine what % it is of last year's sales.

They sold 195 – 178 = 17 fewer books
this is 17 ÷ 195 = 8.7%

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice Exercises

1) Change these decimal numbers to percents. Round to 2 places.

a) 0.035

b) 2.541

c) 0.1357

d) 12.338

2) Change these fractions to percents. Round to 1 decimal place:

a)

b)

c)

d)

3) Change these percents to decimal numbers. Round to 4 places.

a) 9.57%

b) 19.153%

c) 0.57%

d) 231.65%

4) Fill in the blanks. Round to 2 places.

a) 13.6% of ___ = 265

b) 6.87% of 24 = ____

c) 56 out of 123 = ___%

5) Round answers to 2 places.

a) Harry bought a pair of shoes on sale at 35% off. If the regular price was $264, how much did Harry pay for the shoes?

b) Maria and 3 friends went out for pizza and sodas. The pizza was $14.75 and each soda cost $1.25. If all 4 girls had a soda:

i) what was the total bill before taxes?
ii) They leave a 15% tip rounded to the nearest 10 cents. Find the amount of the tip.
iii) Find the total bill with the GST (7%) and the PST(8%). (Don't include the tip here)
iv) If the girls shared the cost equally, how much does each one pay for lunch?

(don't forget the tip.)

c) Anne-Marie got a $500 bond for her birthday which pays 6.7% interest over a 10 year term. How much money will her bond earn in interest if she cashes it in after 10 years?

Solutions

1) Change these decimal numbers to percents. Round to 2 places.

a) 0.035 = 3.5%

b) 2.541 = 254.1%

c) 0.1357 = 13.57%

d) 12.338 = 1233.8%

2) Change these fractions to percents. Round to 1 decimal place:

a)

b)

c)

d)

3) Change these percents to decimal numbers. Round to 4 places.

a) 9.57% = 0.0957

b) 19.153% = 0.1953

c) 0.57% = 0.0057

d) 231.65% = 2.3165

4) Fill in the blanks. Round to 2 places.

a) 13.6% of 1948.53 = 265

b) 6.87% of 24 = 1.65

c) 56 out of 123 = 46.53 %

5) Round answers to 2 places.

a) shoes on sale at 35% off. regular price $264

If they were 35% off, he paid 65% of $264 = 0.65 × 264 = $171.60

b) pizza $14.75 and 4 sodas @ $1.25. :

i) total bill before taxes = $14.75 + 4 × $1.25 = $19.75
ii) they leave 15% tip = 0.15 × $19.75 = $2.96 which they round to $3.00
iii) add GST (7%) = 1.07 × $19.75 = $21.13 , add PST(8%) = 1.08 × $21.13 = $22.82.
iv) each one pays ($22.82 + $3) ÷ 4 = 6.455 or $6.46 for lunch? (we didn't forget the tip.)

c) Anne-Marie's bond earns 500 × 0.067 × 10 = $335 in interest after 10 years?

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