Fractions

Multiply or divide both by the same number

HintKeep the fraction worth what it is — change the top and bottom in step, up or down.

The whyDoing the same thing to top and bottom is really multiplying by a disguised 1, such as 2/2, so the fraction's value cannot change.

3/4

HintLook for the biggest number that goes into both 6 and 8, then cancel it.

The why6 and 8 share a common factor of 2. Dividing both by 2 gives 3/4 — the same amount, in lowest terms.

15

HintSplit the amount into four equal parts, then take three of them.

The whyOne quarter of 20 is 5, so three quarters is 3 × 5 = 15. Find one part first, then scale up.

2/3

HintRewrite both over a shared denominator so the pieces are the same size to count.

The whyOver a common denominator of 15, 2/3 = 10/15 and 3/5 = 9/15, so 2/3 is the larger.

0.5

HintThink of it as one whole shared between two — or half of a pound.

The whyA fraction is a division: 1 ÷ 2 = 0.5. Knowing the common ones (1/2, 1/4, 3/4, 1/10) by heart speeds up percentage work later.

5/6

HintTurn both into sixths first, then add the tops.

The whyOver a denominator of 6, 1/2 = 3/6 and 1/3 = 2/6. Add the numerators: 3 + 2 = 5, giving 5/6.

The denominator

HintOnce the pieces are equal in size, you only count how many — the size label is unchanged.

The whyThe denominator names the size of each piece. Adding thirds to thirds still gives thirds — you add how many pieces (the numerators), not the size.

8/15

HintMultiply straight across — tops together, bottoms together.

The whyMultiply the numerators (2 × 4 = 8) and the denominators (3 × 5 = 15) to get 8/15. No common denominator is needed for multiplying.

It gets smaller

HintYou are taking only a part of the number, not the whole of it.

The whyMultiplying by 1/2 means finding half of the number, so the answer is smaller. Multiplying only grows a number when you multiply by more than 1.

12

HintAsk how many halves fit inside six whole ones.

The whyDividing asks 'how many fit?'. Twelve halves fit into 6, which is why dividing by 1/2 is the same as multiplying by 2.

multiply

HintThe reciprocal turns the division into its opposite operation.

The whyFlipping a fraction gives its reciprocal; dividing by 3/4 is the same as multiplying by 4/3 — the same result from an easier sum.

11/4

HintCount how many quarters are in two whole ones, then add the extra three.

The whyTwo wholes are 8 quarters; adding the 3 quarters gives 11 quarters, written 11/4. Multiply the whole by the denominator, then add the numerator.