Fractions

18

HintReduce the known pair to its lowest terms first, then scale that up until its top matches 12.

The why30/45 cancels to 2/3, and 12 is 6 × 2, so the bottom must be 6 × 3 = 18. Cross-multiplying is the quick check: 12 × 45 = 540 and 30 × 18 = 540, so the two fractions really are equal.

3/4

HintFind the highest common factor of both numbers and cancel it in one go, or cancel in stages.

The why36 and 48 share a highest common factor of 12, so dividing both by 12 gives 3/4. Cancelling in stages (÷2, ÷2, ÷3) reaches the same place — keep going until top and bottom share no factor except 1.

64

HintThree equal parts make 48, so find one part first, then build up to four of them.

The whyIf three quarters is 48, one quarter is 48 ÷ 3 = 16, and the whole is four quarters: 4 × 16 = 64. Working backwards from a fraction of an amount is a step up from finding one, and it is the same reasoning used later for reverse percentages.

10/15 and 9/15, so 2/3 is the larger

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

The whyFifteen is the smallest number both 3 and 5 divide into, so fifteenths are the shared piece — once the pieces match, only the counts need comparing.

0.375

HintFind one eighth first by halving a quarter, then take three of them.

The whyA fraction is a division, so 3 ÷ 8 = 0.375. Quicker: 1/4 = 0.25, so 1/8 = 0.125 and 3/8 = 3 × 0.125 = 0.375. Knowing the eighths as well as halves, quarters and tenths by heart speeds up percentage work later.

sixths; 3/6 and 2/6, so the sum is 5/6

HintThe pieces must be the same size before the tops can be counted together.

The whySix is the smallest number both 2 and 3 divide into — the reason halves and thirds can only be added once both are renamed in sixths.

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 and 15, giving 8/15

HintStraight across on each row — no shared denominator is needed for this one.

The whyUnlike adding, multiplying needs no common denominator: you are taking 2/3 of 4/5, so the counts multiply and the piece sizes multiply at the same time.

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

HintWork backwards one step at a time: undo the division first, then undo the halving.

The whyDividing by 2/3 means multiplying by 3/2, so half the number × 3/2 = 9. That makes half the number 9 × 2/3 = 6, and the number itself 12. Checking forwards: half of 12 is 6, and 6 ÷ 2/3 = 6 × 3/2 = 9.

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.

8 and 11 — so 2 3/4 = 11/4

HintEvery whole one hides four quarters — count those first, and the spare quarters join on at the end.

The whyThe quick rule — multiply the whole number by the denominator, then add the numerator — is exactly this quarter-counting written as a recipe.