Number, Year 9: percentages, prime factors and methods

60

HintThe sale price is not the whole of the old price; decide what percentage of it you have been given.

The why£48 is 80% of the original, so 10% is £6 and 100% is £60. Check forwards: 20% of £60 is £12, and £60 − £12 = £48.

The 10% was taken off the original price, not off £90

HintAsk which amount each of the two percentages is a share of.

The whyThe original was £100: 10% of £100 is £10, leaving £90. But 10% of £90 is only £9, so adding it back falls £1 short. A percentage always belongs to the amount it was worked out from.

125% and £12

HintA rise keeps all of what was there before and adds some more on top.

The why100% + 25% = 125%, so £15 ÷ 1.25 gives the old pay. Check forwards: 25% of £12 is £3, and £12 + £3 = £15.

£8000

HintScale the deposit down to a small, convenient percentage first, then build up to the whole.

The why15% is £1200, so 5% is £400 and 100% is 20 × £400. Whenever you know what a percentage of an amount is worth, you can scale to 100% to recover the original.

Take 30% off £50 and see whether it gives £35

HintRun the story forwards again, starting from the price she found.

The why30% of £50 is £15, and £50 − £15 = £35, so she is right. Running the change forwards is the quickest way to catch the common mistake of adding the percentage back on to the sale price.

12

HintKeep only the primes found in both lists, each as many times as both can supply.

The whyBoth numbers contain two 2s and one 3, so the HCF is 2 × 2 × 3. The third 2, the second 3 and the 5 are each missing from one of the numbers, so they cannot be part of a common factor.

144

HintEach prime must appear often enough to cover whichever of the two numbers needs it most.

The whyA common multiple must contain all of 36 and all of 48, so it needs four 2s (for 48) and two 3s (for 36): 16 × 9. With any fewer, one of the numbers would not divide into it.

60 and 84

HintEach number owns its own region together with the shared middle.

The whyThe left circle holds 5 × 2 × 2 × 3 = 60 and the right circle holds 2 × 2 × 3 × 7 = 84. The overlap alone gives the HCF, 12, and every prime in the diagram multiplied together gives the LCM, 420.

lowest common multiple

HintEverything in each circle is present, so each of the two numbers divides into the result.

The whyThe whole diagram contains every prime of the first number and every prime of the second, with the shared ones counted once. That is exactly enough for both numbers to divide into it, and no more.

A common factor has to divide into both numbers

HintThink what would go wrong if you included a 5 when only one of them contains a 5.

The whyA prime that is missing from one number cannot be part of anything that divides it. So the HCF is built only from shared primes, while the LCM, which both numbers must divide into, needs every prime that either number has.

900

HintTwenty-five is a quarter of a hundred.

The why36 ÷ 4 = 9, and 9 × 100 = 900. Spotting that 25 is a quarter of 100 turns a long multiplication into two easy steps, quicker and safer than reaching for a calculator.

5 × 20 = 100

HintRound each to its leading digit before multiplying.

The whyThe exact answer is 103.731, so the estimate is close enough to catch a misplaced decimal point or a wrong key press. An estimate run alongside a calculation costs a few seconds.

3 × 50 is about 150, so 1536 is ten times too big

HintRound each value to something easy and multiply those.

The why3.2 × 48 is 153.6. A display of 1536 is the classic sign of a missed decimal point: 32 × 48 was keyed in. A ten-second estimate catches errors that the calculator cannot.

386 ÷ 17

HintTwo of the three need only a halving or a shift in place value.

The whyHalving and multiplying by 100 are one-step mental jobs. Dividing by 17 has no shortcut and does not come out exactly, so a calculator, with an estimate of about 20 alongside, is the sensible tool.

5 coaches

HintCan you book part of a coach, and can anyone be left behind?

The whyFour coaches seat only 208, leaving 26 pupils without a seat, so a fifth is needed. A calculator gives the number; you decide what it means in the situation.