Formulas · Mathematics

Numbers and fractions

5 formulas, each with worked examples. Revision sheet · Practise these formulas

Foundation

Adding two fractions

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

What each letter means
$a$	the top (numerator) of the first fraction
$b$	the bottom (denominator) of the first fraction, not 0
$c$	the top (numerator) of the second fraction
$d$	the bottom (denominator) of the second fraction, not 0

Fractions can only be added when they count the same size of piece. Multiplying each fraction's top and bottom by the other's bottom turns both into pieces of size 1 ÷ (b × d), and then the tops just add.

Why it works

a/b is the same as (a × d)/(b × d), and c/d is the same as (c × b)/(d × b), because multiplying top and bottom by the same number does not change a fraction. Now both count pieces of size 1/(bd), so the counts add to ad + bc.

When to use it

Adding (or, with a minus, subtracting) any two fractions, especially when the bottoms are different.

How to use it

Multiply the first top by the second bottom (ad).
Multiply the second top by the first bottom (bc).
Add those two for the new top; multiply the two bottoms for the new bottom.
Simplify: divide top and bottom by any common factor.

To remember: Cross multiply for the top, multiply the bottoms for the bottom.

Other forms

Special case $\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}$ When the bottoms are already the same (b = d): just add the tops.Brahmagupta gave this rule first.
Same formula, another form $\frac{a}{b} - \frac{c}{d} = \frac{a d - b c}{b d}$ Subtracting works the same way, with a minus between the two products.

Worked examples

Story

Asha eats 1/3 of a pizza and Ravi eats 2/5 of it. What fraction of the pizza did they eat together?

$\frac{1}{3} + \frac{2}{5} = \frac{1 \times 5 + 3 \times 2}{3 \times 5}$
$= \frac{5 + 6}{15} = \frac{11}{15}$
Together they ate 11/15 of the pizza, so 4/15 is left.

Answer: 11/15

Picture

A rectangle is 3/4 m long and 2/3 m wide. Find its perimeter.

$\frac{3}{4} + \frac{2}{3} = \frac{3 \times 3 + 4 \times 2}{4 \times 3} = \frac{17}{12}$
The perimeter is twice (length + width).
$2 \times \frac{17}{12} = \frac{17}{6}$

Answer: 17/6 m, which is 2 5/6 m

Direct

Work out 2/3 + 1/4.

$\frac{2}{3} + \frac{1}{4} = \frac{2 \times 4 + 3 \times 1}{3 \times 4}$
$= \frac{8 + 3}{12} = \frac{11}{12}$

Answer: 11/12

Reverse

What must be added to 2/7 to get 1/2?

Try it first, then show the working

The missing fraction is 1/2 − 2/7.
$\frac{1}{2} - \frac{2}{7} = \frac{1 \times 7 - 2 \times 2}{2 \times 7} = \frac{3}{14}$
Check: 2/7 + 3/14 = 4/14 + 3/14 = 7/14 = 1/2.

Answer: 3/14

Exam

Show that 1/2 + 1/3 + 1/6 = 1, and find 5/6 + (−3/8).

Try it first, then show the working

$\frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6} = \frac{5}{6}$
$\frac{5}{6} + \frac{1}{6} = \frac{6}{6} = 1$
$\frac{5}{6} + \frac{- 3}{8} = \frac{5 \times 8 + 6 \times ( - 3 )}{48} = \frac{22}{48} = \frac{11}{24}$

Answer: 1, and 11/24

Common mistake: Adding the tops and adding the bottoms (1/2 + 1/3 is not 2/5). The bottoms name the size of the pieces, and sizes do not add.

Foundation

Dividing by a fraction

\frac{a}{b} \div \frac{c}{d} = \frac{a d}{b c}

What each letter means
$a$	the top of the fraction being divided
$b$	the bottom of the fraction being divided, not 0
$c$	the top of the fraction you divide by, not 0
$d$	the bottom of the fraction you divide by, not 0

Dividing by a fraction is the same as multiplying by that fraction turned upside down (its reciprocal).

Why it works

Dividing asks "how many lots of c/d fit into a/b?". Multiplying c/d by d/c gives 1, so multiplying both parts of the division by d/c leaves the answer the same while turning the divisor into 1; what is left is a/b × d/c.

When to use it

Any division where the divisor is a fraction, such as how many pieces of a given length fit into a length, or finding a side from an area.

How to use it

Keep the first fraction as it is.
Turn the division into a multiplication.
Flip the second fraction (d/c).
Multiply tops together and bottoms together, then simplify.

To remember: Keep, change, flip: keep the first, change ÷ to ×, flip the second.

Other forms

Same formula, another form $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ The same rule as multiplying by the reciprocal.
Special case $a \div \frac{c}{d} = \frac{a d}{c}$ When dividing a whole number a (a/1) by a fraction.

Worked examples

Story

A 6 m ribbon is cut into pieces 3/4 m long. How many pieces are there?

$6 \div \frac{3}{4} = 6 \times \frac{4}{3}$
$= \frac{24}{3} = 8$

Answer: 8

Picture

A rectangle has an area of 5/6 m² and a length of 5/4 m. How wide is it?

Width = area ÷ length.
$\frac{5}{6} \div \frac{5}{4} = \frac{5}{6} \times \frac{4}{5}$
$= \frac{20}{30} = \frac{2}{3}$

Answer: 2/3 m

Direct

Work out 3/4 ÷ 2/5.

$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}$
$= \frac{15}{8}$

Answer: 15/8, which is 1 7/8

Reverse

A number divided by 2/3 gives 9/4. What is the number?

Try it first, then show the working

The number is 9/4 × 2/3, because dividing by 2/3 undoes multiplying by 2/3.
$\frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = \frac{3}{2}$
Check: 3/2 ÷ 2/3 = 3/2 × 3/2 = 9/4.

Answer: 3/2

Exam

Simplify (2/3 ÷ 4/9) ÷ 3/2.

Try it first, then show the working

$\frac{2}{3} \div \frac{4}{9} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$
$\frac{3}{2} \div \frac{3}{2} = 1$

Answer: 1

Common mistake: Flipping the first fraction instead of the second, or flipping both.

Class 9

A number halfway between two numbers

m = \frac{a + b}{2}

What each letter means
$m$	the number exactly halfway between a and b
$a$	the first number
$b$	the second number

The average of two numbers sits exactly halfway between them on the number line, so between any two different rational numbers there is always another one.

Why it works

Halfway from a to b means going half of the gap: a + (b − a) ÷ 2, which simplifies to (a + b) ÷ 2. When a and b are rational, their sum is rational and so is half of it; and since it is strictly between them, you can repeat forever. That is why the rational numbers are called dense.

When to use it

You need a number between two others (a rational number between two fractions or decimals), or the point halfway between two marks on a number line or a ruler.

How to use it

Add the two numbers.
Halve the sum.
To find more numbers between them, take the average of the new number with either end, and repeat.

To remember: Halfway means average.

Other forms

Same formula, another form $m = a + \frac{b - a}{2}$ Start at a and go half of the gap to b.
Rearranged $b = 2 m - a$ The other end, when you know one end and the halfway point.

Worked examples

Story

Two friends measure the same pencil as 14.2 cm and 14.3 cm. They agree to use the number exactly halfway between. What is it?

$m = \frac{14.2 + 14.3}{2}$
$= \frac{28.5}{2} = 14.25$
They use 14.25 cm.

Answer: 14.25

Picture

On a number line, which point is exactly halfway between −4 and 3?

$m = \frac{- 4 + 3}{2} = \frac{- 1}{2}$
The point is −1/2: 3.5 units from −4 and 3.5 units from 3.

Answer: −1/2

Direct

Find a rational number exactly halfway between 1/3 and 1/2.

$\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$
$\frac{5}{6} \div 2 = \frac{5}{12}$

Answer: 5/12

Reverse

The number halfway between 2/5 and some number b is 1/2. Find b.

Try it first, then show the working

$\frac{2}{5} + b = 2 \times \frac{1}{2} = 1$
$b = 1 - \frac{2}{5} = \frac{3}{5}$
Check: (2/5 + 3/5) ÷ 2 = 1 ÷ 2 = 1/2.

Answer: 3/5

Exam

Find three rational numbers between 3.1415 and 3.1416.

Try it first, then show the working

$\frac{3.1415 + 3.1416}{2} = 3.14155$
$\frac{3.1415 + 3.14155}{2} = 3.141525$
$\frac{3.14155 + 3.1416}{2} = 3.141575$
So 3.141525, 3.14155 and 3.141575 all lie between them (and there are infinitely many more).

Answer: 3.141525, 3.14155 and 3.141575

Common mistake: Halving only one of the numbers, or, with fractions, adding tops and bottoms separately instead of adding the fractions properly.

Class 9

A repeating decimal as a fraction

x = \frac{N}{1 0 ^{n} - 1}

What each letter means
$x$	the repeating decimal 0.NNN…, where the block N starts repeating straight after the point
$N$	the repeating block, read as a whole number (for 0.454545… it is 45)
$n$	how many digits the block has (for 45 it is 2)

A decimal whose block repeats forever is always a fraction. For a block that starts right after the point, the fraction is the block over as many 9s as the block has digits.

Why it works

Multiplying by 10ⁿ moves exactly one block in front of the point, and the endless tail stays the same. Subtracting the original removes the tail: 10ⁿx − x = N, so (10ⁿ − 1)x = N. And 10ⁿ − 1 is n nines: 9, 99, 999, …

When to use it

Changing a pure repeating decimal (the block starts straight after the point) into p/q form, or checking that a repeating decimal is rational.

How to use it

Write x = the decimal.
Count the digits in the repeating block (n) and multiply by 10ⁿ.
Subtract the first line from the second: the tails cancel.
Solve for x and simplify the fraction.

To remember: The block over that many nines.

Other forms

Same formula, another formWhen some digits come before the block (a general repeating decimal), multiply once to move those digits in front of the point, and once more to move one full block too, then subtract. For 2.35777…, 1000x − 100x = 2357.7… − 235.7…, so 900x = 2122 and x = 1061/450.
Special case $x = \frac{d}{9}$ When one digit d repeats (n = 1): 0.777… is 7/9.

Worked examples

Story

A calculator shows 0.272727… as the part of a class that comes to school by bus. What fraction of the class is that?

The block is 27, with 2 digits.
$x = \frac{27}{99} = \frac{3}{11}$
So 3 students in every 11 come by bus.

Answer: 3/11

Picture

A square is split into 9 equal small squares and 3 are shaded. As a decimal the shaded part is 0.333…. Show that this is the same as 3 of the 9 squares.

Let x = 0.333…; one digit repeats, so multiply by 10: 10x = 3.333…
$10 x - x = 3$
$x = \frac{3}{9} = \frac{1}{3}$
That is 3 of the 9 small squares, one third of the big square.

Answer: 1/3

Direct

Write 0.454545… (the block 45 repeating) as a fraction.

Let x = 0.4545…; the block 45 has 2 digits, so multiply by 100: 100x = 45.4545…
$100 x - x = 45$
$99 x = 45$
$x = \frac{45}{99} = \frac{5}{11}$

Answer: 5/11

Reverse

The fraction 4/33 is a repeating decimal whose block has 2 digits. What is the block?

Try it first, then show the working

x = N ÷ 99, so N = 99 × x.
$99 \times \frac{4}{33} = 12$
So 4/33 = 0.121212…

Answer: 12

Exam

Write 2.35777… (the digit 7 repeating) as a fraction in its simplest form.

Try it first, then show the working

Let x = 2.35777…. Two digits (35) come before the block, so multiply by 100: 100x = 235.777…
One digit repeats, so multiply by 10 more: 1000x = 2357.777…
$1000 x - 100 x = 2357 - 235$
$900 x = 2122$
$x = \frac{2122}{900} = \frac{1061}{450}$

Answer: 1061/450

Common mistake: Multiplying by 10 when the block has 2 or more digits (the tails then do not line up, so they do not cancel), or forgetting to simplify.

Class 10

HCF × LCM = the product of two numbers

L = \frac{ab}{H}

What each letter means
$L$	the LCM (lowest common multiple) of a and b
$H$	the HCF (highest common factor) of a and b
$a$	the first positive whole number
$b$	the second positive whole number

For two numbers, the HCF times the LCM equals the two numbers multiplied together. So once you know the HCF, the LCM is the product divided by it.

Why it works

Write both numbers as products of primes. The HCF takes the smaller power of each prime and the LCM the larger. For every prime, smaller power + larger power = the two powers together, so HCF × LCM contains each prime exactly as often as a × b does.

When to use it

Finding the LCM once you have the HCF (or the HCF from the LCM), or finding a missing number from the other number, the HCF and the LCM. Only for two numbers.

How to use it

Find the HCF of the two numbers.
Multiply the two numbers.
Divide the product by the HCF: that is the LCM.

To remember: HCF times LCM is the product of the two.

Other forms

Rearranged $H = \frac{ab}{L}$ The HCF, from the LCM.
Same formula, another formHCF × LCM = a × b, the form the book states.
Special case $L = ab$ When a and b have no common factor except 1 (they are co-prime, H = 1).

Worked examples

Story

One bell rings every 6 minutes and another every 20 minutes. They ring together now. After how many minutes do they next ring together?

They ring together after a common multiple of 6 and 20; the first time is the LCM.
The HCF of 6 and 20 is 2.
$L = \frac{6 \times 20}{2} = 60$

Answer: 60

Picture

Tiles 12 cm long and 18 cm wide are laid, all the same way round, to make a square. What is the side of the smallest such square?

The side must be a multiple of 12 (along the lengths) and of 18 (along the widths): the smallest is the LCM.
$L = \frac{12 \times 18}{6} = \frac{216}{6} = 36$

Answer: 36

Direct

The HCF of 96 and 404 is 4. Find their LCM.

$L = \frac{96 \times 404}{4}$
$= \frac{38784}{4} = 9696$

Answer: 9696

Reverse

The HCF of two numbers is 9 and their LCM is 459. One of the numbers is 27. Find the other.

Try it first, then show the working

$b = \frac{9 \times 459}{27}$
$= \frac{4131}{27} = 153$

Answer: 153

Exam

Can two numbers have 18 as their HCF and 380 as their LCM? Give a reason.

Try it first, then show the working

The HCF divides both numbers, and both numbers divide the LCM, so the HCF must divide the LCM.
380 ÷ 18 = 21 remainder 2, which is not a whole number.
So no two numbers have HCF 18 and LCM 380.

Answer: No: the HCF must divide the LCM, and 18 does not divide 380

Common mistake: Using it for three numbers: 6 × 72 × 120 is not HCF × LCM (the book's Example 4). It holds for two numbers only.