Answer key and worked explanations · P4 · 31 questions · 100 marks · 90 minutes
These answers were worked out here, not copied from an official key. The source paper prints no answers at all, so every answer below is reasoning that can be wrong. Where one looks surprising, re-read the method before telling her she is wrong — the method is printed so it can be checked.
Each question carries the method under Why, and the mistake to watch for under Where she’ll slip. The slip is the useful half: it is what to ask about before she starts writing.
The digit 6 in 63 470 has the same value as __________.
(4) 6 × 10 000
Read the places from the RIGHT: 0 ones, 7 tens, 4 hundreds, 3 thousands, 6 ten-thousands. So the 6 is worth 6 × 10 000 = 60 000.
Counting places from the left, or counting the digit's position ("it is the 1st digit") instead of its place value.
How many sixths are there in 2 wholes?
(2) 12
One whole holds 6 sixths, so 2 wholes hold 2 × 6 = 12 sixths.
Answering 6 — that is how many sixths are in ONE whole, not two.
The following numbers are arranged in descending order: A, 17 898, 8929, 879. A is a 5-digit odd number. What is the value of A?
(4) 71 889
Descending means A is the largest, so A must be GREATER than 17 898 — that rules out 17 791. A must also be odd, so it ends in 1, 3, 5, 7 or 9 — that rules out 18 642 and 27 424. Only 71 889 passes both tests.
Checking only that the number is odd and picking 17 791, forgetting A also has to be bigger than 17 898.
Four different shapes J, K, L and M are shown below. Which of the following figures are symmetrical?
/\ ______
/ \ / \
/ \ / \
/______\ /__________\
J K
________ ____
| | / /
| | / /
|________| /____/
L M
J isosceles triangle K isosceles trapezium
L square M slanted parallelogram(2) J, K and L only
Fold each shape and see whether the two halves land on each other. J (isosceles triangle) folds down the middle. K (isosceles trapezium) folds down the middle. L (square) folds four different ways. M is a slanted parallelogram — there is no fold that makes its halves match. So J, K and L only.
Thinking a parallelogram is symmetrical because it looks balanced. It has ROTATIONAL symmetry (turn it half-way round and it matches) but no LINE of symmetry — and this question asks about folding.
The clock shown below was 45 minutes behind the actual time. To set the clock to the correct time, how many 1/4-turn(s) must the minute hand be moved clockwise?
12
11 1
10 2
|
9 +----> 3 minute hand on 12
hour hand on 3
8 4
7 5
6(3) 3
A full turn of the minute hand is 60 minutes, so a 1/4-turn is 60 ÷ 4 = 15 minutes. The clock is 45 minutes behind, so the minute hand must go forward 45 minutes. 45 ÷ 15 = 3 quarter-turns.
Moving the hand 45 minutes and answering 45, or working with the hour hand instead of the minute hand.
The figure below is made up of 4 identical rectangles. Find the perimeter of the figure.
|<----- 10 cm ----->|
+---+
| |
| |
| |
+-------------------+ |
+---------------+---+---+---------------+ -+- 2 cm
| +-------------------+ -+-
| |
| |
| |
+---+
4 identical rectangles, each 10 cm long and 2 cm wide, set as a
pinwheel: turn the figure a quarter turn and it looks the same.(2) 80 cm
The arrows say each rectangle is 10 cm long and 2 cm wide. The figure is a pinwheel — turn it a quarter turn and it looks exactly the same, so the outline is four identical pieces. One piece is 2 cm (the end of an arm) + 10 cm (the long side of that arm) + 8 cm (the part of the next arm still showing) = 20 cm. Four of them: 4 × 20 = 80 cm.
Adding the four rectangles' perimeters (4 × 24 = 96 cm). That counts the edges where the rectangles touch, and those edges are inside the figure, not on its outline.
Uncle Jack bought 60 boxes of pens. There were 9 pens in each box. He repacked the pens into smaller packets of 5 each. How many packets are there?
(4) 108 packets
First find how many pens there are altogether: 60 × 9 = 540 pens. Then share them into packets of 5: 540 ÷ 5 = 108 packets.
Stopping at 540 (that is the number of PENS, not packets), or dividing 60 by 5 and ignoring the 9 pens per box.
Which of the following is NOT an equivalent fraction of 1/4?
(3) 5/16
Multiply the top and bottom of 1/4 by the same number: 2/8 ✓, 3/12 ✓, 6/24 ✓. For sixteenths you would need 4/16 to make 1/4, so 5/16 is the one that does not fit.
Reading past the word NOT and picking one that IS equivalent. Underline NOT before looking at the options.
Helen was at a restaurant in a zoo. At the restaurant, Helen was facing south-west at first. She then made a 3/4-turn clockwise. Which animal would she be facing then?
Lion N
| ^
Elephant | Tiger |
\ | /
\ | /
Zebra ---- RESTAURANT ---- Leopard
/ | \
/ | \
Giraffe | Monkey
|
Penguin(3) Monkey
Each 1/4-turn is 90°. Going CLOCKWISE from south-west: 1/4-turn → north-west, 2/4-turn → north-east, 3/4-turn → south-east. On the map the south-east path leads to the Monkey.
Turning anticlockwise, which lands on north-west (the Elephant). Check which way the clock hands go before you start counting.
At a bakery, muffins are sold at the prices shown. Nick wants to order 57 muffins for a birthday party. What is the least amount of money he will need to pay for the muffins?
+-------------------------------------------------+ | Muffins on Sale! | | | | 1 for $1.40 6 for $5.70 10 for $8.90 | +-------------------------------------------------+
(2) $51.60
Work out the value of each deal first: 10 for $8.90 is 89c each, 6 for $5.70 is 95c each, and a single is $1.40. So take as many 10-packs as possible: 5 × 10 = 50 muffins for 5 × $8.90 = $44.50. The remaining 7 are cheapest as one 6-pack plus one single: $5.70 + $1.40 = $7.10. Total = $44.50 + $7.10 = $51.60.
Buying six 10-packs (60 muffins for $53.40) or paying for the last 7 as singles ($9.80). Both are more than $51.60 — always price the leftover two ways.
Using all the digits 8, 0, 9, 2, form the smallest multiple of 5.
(1) 2890
A multiple of 5 ends in 0 or 5. There is no 5 among the digits, so the number must end in 0. That leaves 8, 9 and 2 for the first three places, and the smallest arrangement of those is 2, 8, 9. So the number is 2890.
Forgetting that ALL four digits must be used, or trying to put the 0 first to make the number small.
The table below shows the 3-h PSI readings from 8 am to 12 noon on 22nd February. Which one of the line graphs best represents the information in the table?
Time | 8 am | 9 am | 10 am | 11 am | 12 noon ----------+------+------+-------+-------+-------- 3-h PSI | 35 | 55 | 50 | 45 | 40 (1) starts high, dips, rises, then falls (2) starts low, jumps to the peak at 9 am, then falls every step (3) rises to a peak at 10 am, then falls (4) starts low, peaks at 9 am, falls, rises again at 11 am, then falls
(2) Graph (2)
Describe the numbers before looking at the graphs: 35 → 55 is a sharp RISE, then 55 → 50 → 45 → 40 falls every single step. Only graph (2) starts low, jumps to its highest point at 9 am, and then goes down at every point after.
Picking (4), which rises again at 11 am. The table never rises after 9 am — checking the LAST few points is what separates (2) from (4).
Shaun ran 3 km on Monday. He ran 2/5 km more on Monday than on Tuesday. How many kilometres did he run on both days?
(3) 5 3/5 km
Monday = 3 km. He ran 2/5 km MORE on Monday, so Tuesday is the smaller one: 3 − 2/5 = 2 3/5 km. Both days together = 3 + 2 3/5 = 5 3/5 km.
Adding 2/5 to Monday instead of subtracting (giving 6 2/5), or answering 2 3/5 — that is Tuesday alone, and the question asked for BOTH days.
Susan and Tina had the same number of beads at first. After Susan threw away 18 beads and Tina bought another 72 beads, Tina had 4 times as many beads as Susan. How many beads did Susan have at first?
(1) 48 beads
Draw Susan's beads at the end as 1 unit and Tina's as 4 units. They started equal, then Susan lost 18 and Tina gained 72, so the gap between them grew by 18 + 72 = 90 beads. That gap is 4 units − 1 unit = 3 units, so 1 unit = 90 ÷ 3 = 30 beads. Susan has 30 left, and she threw away 18, so she started with 30 + 18 = 48 beads.
Answering 30 — that is what Susan has LEFT at the end, not what she started with. Read the question again before writing the number down.
Among Jim and two other friends, John is the lightest and Jack is the heaviest. The table below shows the total weight of two boys weighing themselves each time. Find Jim's weight.
1st reading | 56 kg 2nd reading | 70 kg 3rd reading | 54 kg John < Jim < Jack
(2) 34 kg
The three readings are the three possible pairs. Since John is lightest and Jack is heaviest, the LIGHTEST pair is John + Jim = 54 and the HEAVIEST pair is Jim + Jack = 70, leaving John + Jack = 56. Adding all three readings counts every boy exactly twice: 54 + 56 + 70 = 180, so the three boys together weigh 90 kg. Jim = 90 − (John + Jack) = 90 − 56 = 34 kg.
Assuming the readings are listed in a helpful order (that the 1st reading is John + Jim). Sort them smallest to largest first — it is the smallest and largest pairs that the clue pins down.
The table shows the number of customers at a restaurant over 4 days; part of it was covered by a stain. The number of customers on Day 3 is equal to the total number of customers on Day 1 and Day 2. The total number of customers on all 4 days is 240 when rounded to the nearest ten. What is the greatest possible number of customers on Day 4?
Day | Number of customers ----+-------------------- 1 | 30 2 | 45 3 | ##### stain ##### 4 | ##### stain #####
94
Day 3 = Day 1 + Day 2 = 30 + 45 = 75. So Days 1, 2 and 3 come to 30 + 45 + 75 = 150. The 4-day total rounds to 240 to the nearest ten, so the real total is anywhere from 235 to 244. The question wants the GREATEST Day 4, so take the greatest total: 244 − 150 = 94 customers.
Taking the total as exactly 240 and answering 90. "Rounds to 240" means a range (235 to 244), and the largest number in that range is what makes Day 4 largest.
The figure is made up of 2 identical squares and 3 identical rectangles. What is the length of the unknown side?
+-----------+ -+
| | |
| square | |
| | |
|<-------- ? --------->| | |
+-----------+----------+-----------+ |
| | rectangle | 18 cm
| +----------------------+ |
| square | rectangle | |
| +----------------------+ |
| | rectangle | |
+-----------+----------------------+ -+
|<------------ 26 cm ------------->|17
Look down the right-hand side: 18 cm is the top square's side plus the three stacked rectangles. Now look at the left: the bottom square sits beside those same three rectangles, so the three rectangles stacked are exactly one square's side tall. That makes 18 cm = one square + one square = 2 squares, so each square has side 9 cm. Along the bottom the whole figure is 26 cm, and the top square (9 cm wide) is lined up at the right-hand end, so the marked length = 26 − 9 = 17 cm.
Forgetting the two squares are IDENTICAL. That fact is what lets 18 be split as 9 + 9 — without it there is nothing to pin the size down.
The table shows the programme for the Children's Day celebration. What was the total duration of the Form Teachers' performances?
07 30 - 07 45 | Flag-raising ceremony 07 45 - 07 55 | Principal's speech 07 55 - 08 00 | Mass Singing 08 00 - 08 10 | Primary 4 Form Teachers' performance (Story-telling) 08 10 - 08 25 | Primary 5 Form Teachers' performance (Skit) 08 25 - 08 40 | Mass Singing 08 40 - 08 50 | Primary 6 Form Teachers' performance (Dance) 08 50 | End of concert
35
Pick out only the rows that say Form Teachers' performance. Primary 4: 08 00 to 08 10 = 10 min. Primary 5: 08 10 to 08 25 = 15 min. Primary 6: 08 40 to 08 50 = 10 min. Total = 10 + 15 + 10 = 35 min.
Including the Mass Singing from 08 25 to 08 40 because it sits between two performances, giving 50 min. It is not a Form Teachers' performance — the rows must be chosen by NAME, not by position.
A farmer planted some trees in a straight row, at an equal distance apart from one another. The distance between the 2nd tree and the 5th tree was 2340 m. What was the distance between the 1st tree and the 10th tree?
T1 T2 T3 T4 T5 ... T10
|<--------------->|
2340 m
the trees are equally spaced
(count the SPACES between them, not the trees)7020
Count GAPS, not trees. From the 2nd tree to the 5th there are 5 − 2 = 3 gaps, so one gap = 2340 ÷ 3 = 780 m. From the 1st tree to the 10th there are 10 − 1 = 9 gaps, so the distance = 9 × 780 = 7020 m.
Counting trees instead of gaps — using 4 trees and 10 trees instead of 3 gaps and 9 gaps. Mark the gaps on a quick sketch before dividing.
Barry has a garden with an area of 216 m². It is made up of a rectangle and a square. The area of the rectangle is 5 times the area of the square. Barry wants to build a fence around part of his garden as indicated by the dashes in the figure. Given that the breadth of the rectangle is 9 m, how many metres of fence does he need?
- - - - - - - - - - - - - - - - -+
' ' | 9 m
' rectangle ' |
- - - - - - - -+------+- - - - - -+
' '
' squ. '
' '
+- - - +
the dashed line is the fence (the whole outline)70
The square is 1 share and the rectangle is 5 shares, so the garden is 6 equal shares: 216 ÷ 6 = 36 m² for the square and 5 × 36 = 180 m² for the rectangle. A square of area 36 m² has side 6 m. The rectangle has area 180 m² and breadth 9 m, so its length = 180 ÷ 9 = 20 m. Now walk the outline: 20 (top) + 9 (right side) + 9 (left side) + 14 (the bottom of the rectangle, which is 20 minus the 6 m the square covers) + 6 + 6 + 6 (the square's other three sides) = 70 m.
Adding the two shapes' perimeters separately (2×(20+9) + 4×6 = 82 m). Where the square joins the rectangle there is no fence — that edge is inside the garden.
At first, Mary had $166 and Larry had $304. Each of them bought 5 similar plates, and each plate was the same price. After buying the plates, Larry had 4 times as much money as Mary had left. What was the price of one plate?
24
They spend exactly the same amount, so the GAP between them never changes: 304 − 166 = $138, before and after. At the end Larry has 4 units and Mary has 1 unit, so the gap is 4 − 1 = 3 units. 3 units = $138, so 1 unit = $46 — that is Mary's money left. Mary spent 166 − 46 = $120 on 5 plates, so one plate costs 120 ÷ 5 = $24.
Not noticing that the difference stays the same when both people spend the same amount. That one idea turns a hard problem into two divisions.
At a bento shop in Tokyo, three friends ordered: Hana bought 4 bento sets and 2 bottles of tea for $35. Ken bought 2 bento sets, 3 mochi cakes and 1 bottle of tea for $35.50. Miki bought 2 bento sets, 1 mochi cake and 2 bottles of tea for $27.50. 2 mochi cakes cost the same as 3 bottles of tea. What is the cost of 1 bento set?
6.75
First swap every mochi for tea: 2 mochi = 3 tea, so 1 mochi = 1½ tea. Ken becomes 2 bento + (3 × 1½) tea + 1 tea = 2 bento + 5½ tea = $35.50. Miki becomes 2 bento + 1½ tea + 2 tea = 2 bento + 3½ tea = $27.50. Both have 2 bento sets, so subtracting cancels them: 2 tea = $35.50 − $27.50 = $8, so 1 tea = $4. Put that back into Miki: 2 bento + 3½ × $4 = $27.50 → 2 bento = $27.50 − $14 = $13.50 → 1 bento = $6.75. (Check with Hana: 4 × 6.75 + 2 × 4 = 27 + 8 = $35 ✓)
Starting from Hana's line. Ken's and Miki's both contain 2 bento sets, so subtracting THOSE two removes the bento and leaves only tea — choosing which two lines to subtract is the whole trick.
Joshua had more than 10 and fewer than 40 candies. When she packed the candies in bags of 7, she was left with 2 candies. When she packed them in bags of 3, she had no candies left. How many candies did Joshua have?
30
"Bags of 7 with 2 left over" means the number is 2 more than a multiple of 7: 7+2=9, 14+2=16, 21+2=23, 28+2=30, 35+2=37. Keeping only those between 10 and 40: 16, 23, 30, 37. "Bags of 3 with none left" means a multiple of 3, and of those four only 30 is. So Joshua had 30 candies.
Listing the multiples of 7 themselves (14, 21, 28, 35) and forgetting to add the remainder of 2.
The figure below shows Marcus's backyard with 2 identical 8-m square ponds. What is the remaining area of the empty space around the two ponds in the backyard?
|<-------------- 20 m -------------->| +------------------------------------+ -+ | | | | +----------+ +----------+ | | | | pond | | pond | | | | | 8 m sq | | 8 m sq | | 15 m | +----------+ +----------+ | | | | | +------------------------------------+ -+
172
Find the whole backyard first: 20 × 15 = 300 m². Each pond is an 8 m SQUARE, so its area is 8 × 8 = 64 m², and there are two: 2 × 64 = 128 m². The empty space is what is left: 300 − 128 = 172 m².
Treating "8-m square pond" as an area of 8 m² instead of a square with side 8 m, or subtracting only one pond.
At a bakery, the price of a cupcake was $2 and the price of a tart was $5. Mrs Lim paid $38 to buy a total of 13 cupcakes and tarts. How many more cupcakes than tarts did Mrs Lim buy?
5
Suppose all 13 were cupcakes: that would cost 13 × $2 = $26. She actually paid $38, which is $12 more. Changing one cupcake into a tart adds $5 − $2 = $3 to the bill, so the number of tarts is 12 ÷ 3 = 4. Then cupcakes = 13 − 4 = 9. The question asks how many MORE: 9 − 4 = 5.
Answering 9 — that is the number of cupcakes, not the DIFFERENCE the question asked for. Circle the words "how many more" before you start.
After a rectangular piece of paper was cut into a maximum of 18 squares of sides 4 cm each, an L-shaped strip of paper was left over as shown. Given that the length of the rectangular piece of paper is 26 cm, what is the perimeter of the rectangular piece of paper before it was cut?
+--+------------------------------+ | | | | | | | | the 18 squares of 4 cm | | | | | | | +--+------------------------------+ | strip | -+- 1 cm +---------------------------------+ -+- |<------------ 26 cm ------------>| the leftover L-strip is the narrow band down the left edge plus the 1 cm band along the bottom
78
Fit the 4 cm squares along the 26 cm length: 26 ÷ 4 = 6 remainder 2, so 6 squares fit across and a 2 cm strip is wasted down one side. With 18 squares in rows of 6, there must be 18 ÷ 6 = 3 rows, using 3 × 4 = 12 cm of the breadth. The leftover strip along the bottom is 1 cm, so the breadth is 12 + 1 = 13 cm. Perimeter = 2 × (26 + 13) = 2 × 39 = 78 cm.
Trying to get the breadth by dividing the squares' area (18 × 16 = 288 cm²) by 26. The L-strip is wasted paper, so the squares' area is NOT the paper's area.
A tortoise can crawl 8 m in one minute. A rabbit can run at a speed of 40 m in one minute. The two are having a race one day. While the tortoise crawls continually, the rabbit rests for 3 minutes after every 2 minutes' run. How far does the rabbit run when the tortoise crawls 360 m?
720
The tortoise never stops, so use it as the clock: 360 ÷ 8 = 45 minutes for the whole race. The rabbit's pattern repeats every 2 + 3 = 5 minutes, and in each 5-minute cycle it covers 2 × 40 = 80 m. In 45 minutes there are 45 ÷ 5 = 9 complete cycles, so the rabbit runs 9 × 80 = 720 m.
Working out 45 minutes × 40 m = 1800 m and forgetting the rabbit spends 3 of every 5 minutes resting. Find the TIME first, then the repeating cycle.
The figure is not drawn to scale. It is made up of two identical rectangles overlapping each other, forming Square A. The area of Square A is 9 cm² and the area of each rectangle is 50 cm². The length of the rectangle is twice its breadth. Find the perimeter of the figure.
+---------+
| |
| |
| |
+------------+---+ |
| | A | |
| +---+-----+
| |
+----------------+
Square A is where the two rectangles overlap48
Square A has area 9 cm², so its side is 3 cm. Each rectangle has area 50 cm² with length = 2 × breadth, so breadth × 2 × breadth = 50 → breadth × breadth = 25 → breadth = 5 cm and length = 10 cm. One rectangle's perimeter is 2 × (10 + 5) = 30 cm, so the two separately total 60 cm. But where they overlap, two sides of Square A are hidden inside each rectangle — four 3 cm edges in all. Perimeter of the figure = 60 − 4 × 3 = 60 − 12 = 48 cm.
Taking Square A's side as 9 cm instead of 3 cm (9 is the AREA), or answering 60 by forgetting that the overlap hides edges.
Mary had 48 stalks of flowers in a basket. 1/3 of them were roses and the rest were lilies and daisies. There were 4 more lilies than daisies. How many daisies should Mary buy such that the number of daisies would be 1/2 of the total number of flowers in the basket?
20
Roses = 1/3 of 48 = 16, so lilies and daisies together = 48 − 16 = 32. There are 4 more lilies than daisies, so take the 4 off and share the rest equally: (32 − 4) ÷ 2 = 14 daisies, and 14 + 4 = 18 lilies. Now she buys more daisies. Wanting daisies to be HALF the total is the same as wanting daisies to equal everything else put together — that is roses + lilies = 16 + 18 = 34. She already has 14, so she must buy 34 − 14 = 20 daisies. (Check: 34 daisies out of 48 + 20 = 68 flowers, and 34 is half of 68 ✓)
Taking "half the total" as half of 48 (= 24) and answering 10. The total GROWS with every daisy she buys, which is why it is easier to match the daisies against everything else.
John divided the corridor of a school building into equal parts of length 4 m, and placed 2 potted plants in each part. For the same corridor, he divided it into equal parts of length 6 m and hung 5 lanterns in each part. If there were 24 more lanterns than potted plants, how long was the corridor?
Figure 1 |<-- 4 m -->| 2 potted plants in each part Figure 2 |<--- 6 m --->| 5 lanterns in each part
72
The two patterns use different part lengths, so compare them over a length both divide — 12 m. In 12 m there are 12 ÷ 4 = 3 parts of plants, giving 3 × 2 = 6 plants, and 12 ÷ 6 = 2 parts of lanterns, giving 2 × 5 = 10 lanterns. That is 10 − 6 = 4 more lanterns for every 12 m. We need 24 more, so the corridor is 24 ÷ 4 = 6 lots of 12 m = 72 m. (Check: 72 m gives 18 × 2 = 36 plants and 12 × 5 = 60 lanterns, and 60 − 36 = 24 ✓)
Comparing 2 plants against 5 lanterns directly and using a difference of 3. The parts are different LENGTHS, so the counts can only be compared over the same distance.
In a restaurant, a rectangular table can seat 10 people and a round table can seat 5 people. During lunch time, all the tables in the restaurant were occupied. The number of each type of table in the restaurant is more than 10 and less than 15. If there were 205 customers, how many rectangular and round tables altogether were there for the 205 customers?
27
"More than 10 and less than 15" means each type is 11, 12, 13 or 14. Seats: 10 × rectangular + 5 × round = 205, and dividing everything by 5 gives 2 × rectangular + round = 41. Now test each value: 11 rectangular → round = 41 − 22 = 19 ✗; 12 → 17 ✗; 13 → 15 ✗ (15 is not less than 15); 14 → 41 − 28 = 13 ✓. So 14 rectangular and 13 round, giving 14 + 13 = 27 tables.
Reading "more than 10 and less than 15" as including 10 and 15, or finding one pair that makes 205 without checking that BOTH counts sit inside the range.