The hardest PSLE 2024 Maths questions

Question 1 of 5

The folded and cut paper

Look at Figure 3 before reading on. The piece marked X has been opened out, and the dashed line across it is the crease. That line is the whole question: X is two triangles, not one.

Why children lost marks. The fold does something most children never notice. The scissors go through two layers, so the piece that comes away is two triangles, not one. Part (b) is where that bites, and the wrong working looks perfectly reasonable all the way down.

What the fold actually does. Folding along AC lifts corner B over and drops it onto the sheet. Call the landing point B′. The flap is now lying on top of the paper. Cut around that flap and you cut both the flap and the triangle of sheet directly underneath it. Open the cut-off piece back out and it is a kite, made of the same triangle twice.

Part (a). Give BD four units. BC is a quarter of BD, so let BC be 1 unit, BD 4 units, and CD the remaining 3 units. Now write down both perimeters without measuring the slanted edge at all.

• X is the kite. Round it: AB is 50 cm, BC is 1 unit, CB′ is 1 unit and B′A is 50 cm, because a fold never changes a length. That is 100 cm + 2 units.
• Y is everything else. Round it: the left edge is 4 units, the bottom edge is 50 cm, then D back up to C is 3 units, then the two cut edges CB′ (1 unit) and B′A (50 cm). That is 100 cm + 8 units.

Both carry the same 100 cm of straight edge, so it cancels out and the slanted edge never has to be worked out at all.

The difference is 8 units − 2 units = 6 units, and we were told that difference is 60 cm.

• 6 units = 60 cm, so 1 unit = 10 cm.
• BD is 4 units, so BD = 4 × 10 = 40 cm.

Part (b). Now the doubling pays. The sheet is 50 cm by 40 cm, so 2000 cm². Triangle ABC has base 50 cm and height BC = 10 cm, so it is 1/2 × 50 × 10 = 250 cm². The kite is that triangle twice.

• Cut away = 2 × 250 = 500 cm².
• Area of Y = 2000 − 500 = 1500 cm².

The near miss to check for. A child who takes away one triangle gets 1750 cm². Every line of their working is right except the first assumption. If that's the number on their paper, they understood the question and missed the fold.

Question 2 of 5

The paper hearts and stars

Why children lost marks. There is no formula here, and the picture doesn't do the work. It shows two boards, not a method. The question asks for the *smallest* number, so your child has to search, and a search only works if it's organised. Most of the lost marks are guesses that stopped at the first thing that nearly worked.

Start by writing what each board ends with. Say some students pinned hearts and the rest pinned stars.

• Hearts: it starts at 22, and every heart student adds 5. So hearts = 22 + 5 × (heart students).
• Stars: it starts at 2, and every star student adds 8. So stars = 2 + 8 × (star students).

The two have to end equal. The gap to close is 22 − 2 = 20 in favour of hearts, and every star student closes it by 8 while every heart student widens it by 5.

Now make a systematic list. This is the method the question is really testing. Step the star pupils one at a time, work out how many extra hearts you'd need to match, and check whether that is a whole number of 5-heart pupils.

The first two rows can't work: 10 and 18 stars are fewer than the 22 hearts already up, so the hearts can never come down to meet them. Rows 3 and 4 need 4 and 12 extra hearts, and neither divides by 5. Row 5 needs 20, and 20 is exactly 4 pupils.

So 5 star students and 4 heart students. Check it: hearts = 22 + 4 × 5 = 42, stars = 2 + 5 × 8 = 42. Equal.

Question 3 of 5

The triangle sitting on the square

Why children lost marks. The 68° isn’t near either angle being asked for, and there’s no obvious first move. The unlock is one line that reads like scene-setting: FB is parallel to ED.

Use the parallel line first. ED is the bottom of the square and AC is its top, so ED and AC are parallel. FB is parallel to ED, so FB is parallel to AC as well. That is what lets an angle at F travel down to A, and it is also the whole of part (c).

Part (a). Look at triangle AFB. Because FB is parallel to AC, ∠FBA and ∠BAC are alternate angles, so ∠FBA = 60°, the corner of the equilateral triangle. Now the triangle is finished: ∠FAB = 180° − 68° − 60° = 52°.

Four angles meet at A and together they go all the way round, so they add to 360°: ∠FAB, ∠BAC, ∠CAE and the one we want.

• ∠FAB = 52°, just found.
• ∠BAC = 60°, a corner of the equilateral triangle.
• ∠CAE = 90°, a corner of the square.
• So ∠FAE = 360° − 52° − 60° − 90° = 158°.

Part (b). BD is the line to think about. Triangle BCD is isosceles, and this is the step children miss. BC is a side of the equilateral triangle, CD is a side of the square, and both are the same length as AC. So BC = CD. Pull that triangle out on its own and it's easier to see.

• At C, ∠BCD is the triangle's 60° plus the square's 90°, so ∠BCD = 150°.
• The other two angles in triangle BCD share what is left: (180° − 150°) ÷ 2 = 15° each. So ∠CBD = 15°.
• BD runs inside angle ABC, so ∠ABD = 60° − 15° = 45°.
• And ∠FBA is the 60° from part (a). So ∠FBD = 60° + 45° = 105°.

Part (c). AFBC has FB parallel to AC, which we sorted out at the start. FA and BC aren't parallel. FA leans down to the left, BC leans down to the right. One pair of parallel sides makes it a trapezium.

Question 4 of 5

The two quarter circles

Why children lost marks. Only two lengths are printed, and neither one is the radius. The figure has to be read before it can be used, and the thing to read isn’t written down anywhere in the question.

The thing to see: the radius IS the rectangle's longer side. Follow the upper-left arc. It starts on the middle line and sweeps round until it lands exactly on the far corner of the lower-left rectangle. So the distance it turns through is that rectangle's long side. Once your child sees that, the rest is short.

Part (a). The rectangle's perimeter is 118 cm, so one long side plus one short side is 118 ÷ 2 = 59 cm. The short side is the 24 cm marked on the figure.

• Longer side = 59 − 24 = 35 cm.
• The radius is that same length, so the radius is 35 cm.

Part (b). Put the two arcs together. The two quarter circles are identical, so side by side they make one half circle of radius 35 cm. A half circle's arc is just π × r, with no halving of anything else.

• The two arcs together: 22/7 × 35 = 110 cm.
• Two long sides of rectangle, one top and one bottom: 35 + 35 = 70 cm.
• Two short sides, one on each side: 24 + 24 = 48 cm.
• The two little gaps on the middle line, top and bottom: 7 + 7 = 14 cm.

Perimeter = 110 + 70 + 48 + 14 = 242 cm.

Worth saying out loud. The 7 cm looks like decoration and it isn't. It's two real pieces of the outline, one at the top and one at the bottom. Leaving them out is the commonest way to land just short of the answer.

Question 5 of 5

The three children and the charity

Why children lost marks. Three fractions, three different amounts, and no picture given. Most children try to draw the money first and get stuck, because the money is exactly what they don't know yet. Start from the donations instead. The question tells you how those compare.

Draw the donations, not the money. Devi and Eric gave the same amount, and Haziq gave 3 times what Eric gave. Call Eric's donation 2 units, so nothing lands on a half later. Then Devi gave 2 units and Haziq gave 6.

Now each donation walks back to the money it came from. Devi's 2 units were a quarter of hers, so her money is 4 lots of 2, which is 8 units. Eric's 2 units were 2/7 of his, so 1 unit is 1/7 and his money is 7 units. Haziq's 6 units were 2/5 of his, so 3 units is 1/5 and his money is 15 units.

Part (a) is now just looking. Haziq's bar is longest, Eric's is shortest. No sum needed. Worth pausing on that one, because it's the part that surprises: Eric gave away exactly as much as Devi, and he's the one who started with least.

Part (b). Count all the units: 8 + 7 + 15 = 30 units, and that's the $1560.

• 1 unit = 1560 ÷ 30 = $52.
• Haziq gave 6 units, so 6 × 52 = $312.

Worth checking, and it's quick. Haziq had 15 × 52 = $780, and 2/5 of 780 is 312. Devi had $416 and gave a quarter, so $104. Eric had $364 and gave 2/7, also $104. Same amount, exactly as the question said.