Rational Numbers

Worked example

1. 1

   $2.35=\frac{235}{100}$ — 2.35 has two decimal places, so it is 235 hundredths. That is already one integer over another.

2. 2

   $\frac{235}{100}=\frac{47}{20}$ — Divide top and bottom by 5 to simplify. Simplest form is neater, and it is still integer over integer.

3. 3

   $3\frac{2}{3}=\frac{3\times 3+2}{3}$ — Turn the mixed number into thirds: 3 wholes are 3 × 3 = 9 thirds, plus the 2 thirds.

4. 4

   $\frac{9+2}{3}=\frac{11}{3}$ — 9 + 2 = 11, so it is 11 over 3.

5. 5

   $0.777\dots =\frac{7}{9}$ — Check it: 7 ÷ 9 = 0.777…. Every repeating decimal can be written as a fraction. The Recurring Decimals lesson shows how to prove it.

6. 6

   $\sqrt{0.49}=0.7$ — 0.7 × 0.7 = 0.49, so the square root comes out exactly. It does not go on for ever.

7. 7

   $0.7=\frac{7}{10}$ — 0.7 is 7 tenths, so this root is one integer over another.

Worked example

1. 1

   $\frac{7}{40}:40=2^3\times 5$ — Write the bottom as primes. 7 and 40 share no factor, so it is already simplified. Only 2s and 5s.

2. 2

   $\frac{7\times 25}{40\times 25}=\frac{175}{1000}$ — Multiply top and bottom by 25, because 40 × 25 = 1000. Since 10 = 2 × 5, a bottom of only 2s and 5s can always be made into 10, 100, 1000…

3. 3

   $\frac{175}{1000}=0.175$ — 175 thousandths is 0.175. It stops, so it is terminating.

4. 4

   $\frac{7}{12}:12=2^2\times 3$ — Now $\frac{7}{12}$. Write 12 as primes: a factor 3 sneaks in. 7 and 12 share no factor, so the 3 cannot cancel.

5. 5

   $\frac{7}{12}=0.58333\dots$ — 10, 100, 1000… are made only of 2s and 5s, so a bottom with a 3 can never become one of them. The 3s go on for ever: recurring.

Worked example

1. 1

   $\frac{1}{3}+\frac{1}{2}=\frac{2}{6}+\frac{3}{6}$ — Add the two numbers. Change both to sixths first, so the bottoms match.

2. 2

   $\frac{2}{6}+\frac{3}{6}=\frac{5}{6}$ — 2 + 3 = 5, so the total is 5 sixths.

3. 3

   $\frac{5}{6}÷2=\frac{5}{12}$ — Halve the total. Half of the sum of two numbers is always exactly halfway between them.

4. 4

   $\frac{4}{12}<\frac{5}{12}<\frac{6}{12}$ — Check with the same bottom: $\frac{1}{3}=\frac{4}{12}$ and $\frac{1}{2}=\frac{6}{12}$. 5 is between 4 and 6, so it fits.

Worked example

1. 1

   $\sqrt{2}<?<\sqrt{3}$ — Squaring keeps positive numbers in order, so your number squared must land between 2 and 3.

2. 2

   $1.4^2=1.96$ — Try 1.4. 1.96 is less than 2, so 1.4 is too small.

3. 3

   ${1.5}^2=2.25$ — Try 1.5. 2.25 is between 2 and 3, so it fits. And $1.5=\frac{15}{10}$, so it is rational.