JMC 2017

Alternate angles

HintOne is on each side of the crossing line.

The whyA line crossing two parallel lines makes alternate (Z) angles equal, corresponding (F) angles equal and co-interior (C) angles that sum to 180°. Competition diagrams hide the Z; look for it whenever two lines are marked parallel.

60

HintThe sale price is four fifths of what it was, so undo that multiplication.

The whyCutting by 20% multiplies by 0.8, so undoing it means dividing: 48 ÷ 0.8 = 60. Adding 20% back on to £48 gives £57.60, which is not the same thing. Multiplier thinking also chains: a 10% rise followed by a 10% fall is × 1.1 × 0.9 = × 0.99, a 1% loss overall. Worth knowing as fractions too: 5% is 1/20, 10% is 1/10, 20% is 1/5, 25% is 1/4.

15

HintAdd all nine numbers, then share the total equally among the rows.

The whyWhatever nine consecutive numbers a magic square uses, the line total is their sum divided by three — for 4 to 12 it is 24. Knowing the line total turns a magic-square question into simple subtraction.

Simplify it fully

HintGet the numbers small before you scale them up again.

The whyCancel 216/480 to 9/20 before finding the same proportion of 800: 9/20 × 800 = 360 is far easier than 216/480 × 800. Common factors are always cheaper to remove early than to carry through a calculation.

45

HintFold a square along its diagonal.

The whyA square cut along a diagonal gives two of them. Whenever two sides from a vertex are equal and the angle between them is a right angle, the other two corners come free; when the angle between equal sides is 60°, the triangle is equilateral instead.

Smaller

HintTry a half: a half of a half.

The why(½)² = ¼, which is less than ½. Squaring makes numbers bigger only above 1; between 0 and 1 it shrinks them, and square-rooting does the opposite (√¼ = ½, bigger). 'Always' statements about squaring are usually false for this reason.

13

HintIt is just above 12, whose square you know is 144.

The whyKnowing squares up to 15² = 225 and cubes up to 6³ = 216 by heart saves time all through the paper. A number divided by its own square root leaves no remainder — the root is a factor.

Sum of their speeds

HintEach rider closes part of the gap every hour.

The whyApproaching each other, the gap closes at the combined speed; travelling the same way, it changes at the difference of the speeds. If one sets off later, first work out how far the other has gone by then, and only then use the combined speed.

The distributive law

HintRead it right-to-left too: it is how you pull a common factor out of two products.

The whyRead it right to left and it becomes a shortcut: 17 × 83 − 17 × 23 = 17 × (83 − 23) = 17 × 60. Whenever two products share a factor, take the factor out before touching the arithmetic.

Their lowest common multiple

HintNot the product — unless the two share no factor at all.

The whyMultiples of both 4 and 9 are the multiples of 36, and there the LCM is the product because 4 and 9 share no factor. Multiples of both 4 and 6 are the multiples of 12, not 24, because 4 and 6 share the factor 2.

6

HintThe cut can pass through every face of the cube at most once.

The whyTriangles, squares, rectangles, pentagons and even regular hexagons can all be cut from a cube — but never a heptagon, since a cube has only six faces to cross. Cross-section questions reward imagining the cut rather than trusting your first instinct.

middle; most common; smallest

HintThree summaries, three different questions about the list.

The whyFor an even-sized list the median is the mean of the two middle values. Check which average a question names before doing anything — the three answer different questions.

Adding them

HintMultiplying by 1 never changes a number at all.

The whyMultiplying by 1 leaves a number exactly as it is, so once one of the numbers involved is 1, addition (or even subtraction) can beat multiplication comfortably. This flips as soon as both numbers grow past 2, e.g. 4 × 5 = 20 easily beats 4 + 5 = 9. Never assume × or ÷ automatically gives the biggest answer — check with the actual numbers involved.

First work out how many minutes reach the next o'clock time

HintSplit the duration into two chunks around that point.

The whyFor instance, adding 1 hour 25 minutes to 9:50 am is easier as 10 minutes to reach 10:00 am, then the remaining 1 hour 15 minutes on top, giving 11:15 am. Splitting a time calculation at the next o'clock reading avoids minute totals over 60, which are easy to miscount.

Convert each one to a decimal or a common form

HintYou can't compare fractions by eye if their denominators differ.

The whyTurning expressions like 1/2 + 1/3 and 3/4 × 2/5 into decimals (0.83 and 0.3) puts them on the same footing, so ordering is simple. Estimating first — is the answer bigger or smaller than 1? — narrows things down before exact working.

Half of it

HintSlide the apex along the side above the base: base and height never change, so neither does the space inside.

The whyA rectangle split along a diagonal always gives two congruent right-angled triangles, each covering exactly half the rectangle's area — so if you know a triangle's area, doubling it gives you the area of the rectangle sharing its base and height, and vice versa.

18

HintA minute is 60 seconds; find three tenths of that.

The why1 minute = 60 seconds, so any fraction of a minute converts to seconds by multiplying that fraction by 60 — one tenth is 6 seconds, three tenths is 18. The same multiply-the-whole-amount idea works for any 'fraction of a unit' conversion, like finding a fraction of an hour in minutes.

Add its digits — if that sum is a multiple of 9, so is the number

HintTry it on 9, 18, 27, 36 — notice a pattern in what they add up to.

The whyThis digit-sum test also works for 3 (digit sum divisible by 3). It's especially handy for spotting multiples of a compound number like 45 = 5 × 9: since 5 and 9 share no common factor, a number is a multiple of 45 exactly when it's a multiple of both 5 and 9 — check the units digit for 5, and the digit sum for 9.

2

HintSplitting the polygon into triangles from one vertex always gives two fewer triangles than sides.

The whyThe interior angles of an n-sided polygon sum to (n − 2) × 180° — a pentagon (5 sides) splits into 3 triangles for 540°, a heptagon (7 sides) splits into 5 triangles for 900°. This formula, together with angles at a point summing to 360°, unlocks most polygon angle-chasing questions.

Also coloured

HintPicture folding the paper along that line — everything marked lands on something else marked.

The whyWith two lines of symmetry (say horizontal and vertical), a coloured square off both lines belongs to a matched set of four — its reflections in each line, and the reflection of those again. This is why symmetric colouring almost always comes in pairs or fours rather than single squares.

Always even

HintTwo times anything is automatically a multiple of two.

The whyDoubling any whole number, however large or however odd it started, produces a multiple of 2 — an even number, with no exceptions. This is a handy check in problems built from a small set of coin, weight or step values: if two equal 'unknown' amounts are combined, their total must be even, which can rule out several possibilities at once.

Pair each even number with the odd number just below it

HintEvery matched couple, like (8 − 7) or (20 − 19), comes to exactly 1.

The whyPairing 8 − 7, 20 − 19, and so on, each pair contributes exactly 1, so the whole calculation collapses to (number of pairs) × 1 — turning a long addition-and-subtraction into simple counting. This trick works whenever two lists run in step with a constant gap between matching terms.

7

HintCheck the last digits of 1² through 9² and see which ones actually turn up.

The whySquares can only end in 0, 1, 4, 5, 6 or 9 — 2, 3, 7 and 8 never appear as a square's last digit. This quick check rules out wrong answers fast: a number ending in 7 can never be a perfect square, whatever else about it looks promising.

Write the perimeter as a multiple of the unknown length, then solve

HintCount how many of that one edge go around the outside before setting up your equation.

The whyIf a large shape's outline is made up of, say, 12 copies of a single edge length x, its perimeter is 12x — so a known total perimeter turns straight into a one-step equation for x. Once x is known, any other perimeter built from the same repeating edge follows immediately.

eliminate; choose

HintSubtracting one equation from the other often removes a variable cleanly.

The whySubtracting a 'total items' equation from a 'total parts' equation (like counting vehicles versus counting their wheels) often leaves a simpler equation in two unknowns, which usually has more than one whole-number solution. The extra worded clue — an inequality between two of the unknowns — is what narrows several candidates down to exactly one.