☰ Year plan · Mathematics

Rational numbers

Week 5 · Rational numbers: their forms and the four operations · 3 lessons
1

Rational numbers as fractions, decimals, recurring decimals and percents

Mathematics Prep 1 Term 1, Unit 1, Lesson 5
GoalSay what a rational number is and write it as a fraction, decimal, recurring decimal or percent.
New words
rational number · a number a/b with a and b integers and b ≠ 0Q · the set of all rational numbersrecurring decimal · a decimal whose digits repeat forever, like 0.3̄percent · a fraction out of 100
Explanation

A rational number is any number you can write as a fraction a/b, using two integers, with the bottom number b different from 0. Q is the set of all rational numbers. Every integer is rational because 7 = 7/1 and −2 = −2/1, so N ⊂ Z ⊂ Q. One rational number has many forms: 3/4 = 0.75 = 75%. To change a fraction into a decimal, divide the top by the bottom; to get a percent, multiply the decimal by 100. Some fractions give a decimal that never ends and repeats, such as 1/3 = 0.3̄, written with a bar over the repeating digit. On a calculator 1/3 shows 0.33333333. The number 7/0 is not a number at all, since b must not be 0.

FractionDecimalPercent
1/20.550%
1/40.2525%
3/40.7575%
1/50.220%
1/30.3̄about 33%
Worked examples
Write 3/4 in three forms. Divide: 3 ÷ 4 = 0.75. Multiply by 100: 0.75 × 100 = 75%. So 3/4 = 0.75 = 75%.
−3−2−10123−1.50.52
Integers and decimals are rational. 7 = 7/1 and −2 = −2/1. The decimal 0.6 = 6/10 = 3/5. The numbers −1.5, 0.5 and 2 all sit on the number line between −3 and 3.
A recurring decimal. 1/3 = 1 ÷ 3 = 0.333… = 0.3̄. The digit 3 repeats forever. Compare 1/4 = 0.25, which stops, so it is a finite decimal.
Class activity

Forms match. With a grown-up, write six cards: 1/2, 0.5, 50%, 1/5, 0.2, 20%. Shuffle and match each fraction with its decimal and percent. Add your own set with 3/10.

Practice
1
Write 3/8 as a decimal and as a percent.
2
Write 0.35 as a fraction in simplest form.
3
Which fraction gives a recurring decimal: 1/4 or 1/3?
4
Is 5 a rational number? Is 7/0?
5
Karim says 1/3 = 0.3 exactly. What is wrong?