3.3 Percentages

You have probably met percentages in various contexts, such as a 3% pay rise, 10% interest on a loan, or 20% off goods in a sale. A percentage is a fraction expressed in hundredths. So 1/2, i.e. one-half, is 50/100, or fifty-hundredths, and we say that this is 50 per cent, which is usually written as 50%. This literally means 50 in every 100. The advantage of using percentages is that we are always talking about hundredths, so percentages are easy to compare, whereas with fractions we can divide the whole thing into arbitrary numbers of parts, eighths, sixteenths, fiftieths, or whatever we choose. It is not immediately obvious that 19/25 is larger than 15/20, but if these fractions are expressed as percentages, i.e. 76% and 75%, respectively, then it is easy to see that the former number is the larger of the two. But how do we convert fractions to percentages, or percentages to fractions?

The way that you convert a fraction into a percentage is by multiplying the fraction by 100%. So to convert 1/2 to a percentage:

one divided by two multiplication 100 percent equals one division two multiplication 100 percent equals 50 percent

Three-quarters, 3/4, converts to:

three divided by four multiplication 100 percent equals three division four multiplication 100 percent equals 75 percent

On occasions, you will need to convert a percentage to a fraction, and to do this you need to remember that a percentage is a fraction expressed in hundredths and then cancel as appropriate.

Thus, 75 percent equation sequence part 1 equals part 2 75 divided by 100 equals part 3 15 divided by 20 equals part 4 three divided by four

(having divided the top and the bottom of the fraction by 5, then by 5 again),

and 15 percent equation sequence part 1 equals part 2 15 divided by 100 equals part 3 three divided by 20

(having divided the top and the bottom by 5).

Question 3.3

Convert the following fractions to percentages:

    • a. seven divided by 10

  • 70%

    (Remember, to convert a fraction to a percentage you multiply it by 100%; so

    seven divided by 10 is equivalent to seven divided by 10 multiplication 100 percent equals 70 percent full stop )

    • b. nine divided by 20

  • 45%

    • c. 13 divided by 25

  • 52%

    • d. 63 divided by 100

  • 63%

    • e. 140 divided by 200

  • 70%

    • f. 30 divided by 20

  • 150%

    {Note that this last answer is greater than 100%, because the fraction 30 divided by 20 (or three divided by two ) is greater than 1.}

Question 3.4

Convert the following percentages to fractions with the smallest possible whole numbers on the top and the bottom:

    • a.60%

  • 60 percent equation sequence part 1 equals part 2 60 divided by 100 equals part 3 six divided by 10 equals part 4 three divided by five

    (Here, we have first divided the top and bottom of the sixty-hundredths by 10, and then by 2. Remember that dividing (or multiplying) both the top and bottom of a fraction by any number will produce an equivalent fraction. It is conventional to express fractions with the smallest possible numbers on the top and bottom.)

    • b.64%

  • 64 percent equation sequence part 1 equals part 2 64 divided by 100 equals part 3 32 divided by 50 equals part 4 16 divided by 25

    (Here we have divided the top and the bottom of 64 divided by 100 by 2 and then 2 again, which is the same as dividing by 4.)

    • c.65%

  • 65 percent equation sequence part 1 equals part 2 65 divided by 100 equals part 3 13 divided by 20

    • d.67%

  • 67 percent equals 67 divided by 100

    (This time, there are no whole numbers by which we can divide both 67 and 100 to produce smaller whole numbers on the top and bottom.)

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3.2 Ratios
Next : 3.4 Calculating fractions and percentages of numbers 
3.4 Calculating fractions and percentages of numbers