Problem Solving — Challenge
A hard problem is rarely beaten by cleverness. It is beaten by a method: understand it, pick a strategy, carry it out, then check.
What these problems cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Net — A 3-D shape unfolded flat.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Diagonal — A segment joining two non-adjacent vertices.
- Adjacent — The side next to that angle which is not the hypotenuse.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
A cinema sells adult tickets at $12 and child tickets at $8. What is the cost in dollars for two adults and three children?
- $40
- $44
- $48
- $52
- $60
Worked answer
$48
Two adults cost 2· 12 = 24 dollars.
Three children cost 3· 8 = 24 dollars.
The total is 24+24 = 48 dollars.
Problem 2
A gardener plants trees in a straight line, 4 metres apart. The first and last trees are 36 metres apart. How many trees are there?
- 8
- 9
- 10
- 11
- 36
Worked answer
10
The gaps between the trees number 364 = 9.
A line of trees has one more tree than it has gaps, like fence posts along a fence.
So there are 9+1 = 10 trees. Answering 9 counts the gaps rather than the trees.
Problem 3
A shop sells pens only in packs of 5 and packs of 8. What is the largest number of pens that cannot be bought exactly?
- 19
- 22
- 25
- 27
- 35
Worked answer
27
A total is possible when it can be written as 5a+8b for whole numbers a and b.
Once five consecutive totals are possible, so is every larger one, because adding another pack of 5 steps forward from one of them.
The run 28,29,30,31,32 is possible: 28=5· 4+8, 29=5+8· 3, 30=5· 6, 31=5· 3+8· 2 and 32=8· 4.
Checking downwards, 27 cannot be made: 27-8=19, 27-16=11 and 27-24=3 are none of them multiples of 5. So 27 is the largest impossible total.
Problem 4
A bus holds 45 passengers. How many buses are needed to carry 200 passengers?
- 4
- 4.4
- 5
- 6
- 45
Worked answer
5
20045≈ 4.44, so four buses are not enough.
Four buses carry 180 passengers, leaving 20 still to travel, who need a fifth bus.
The answer must be a whole number of buses, so 5.
Problem 5
The sum of three consecutive even numbers is 42. What is the largest of them?
- 12
- 14
- 16
- 18
- 20
Worked answer
16
Call the middle number n, so the three numbers are n-2, n and n+2.
Their sum is 3n = 42, so n = 14 and the numbers are 12, 14 and 16.
The largest is 16.
Problem 6
A rope 24 metres long is cut into pieces 3 metres long. How many pieces are there?
- 6
- 7
- 8
- 9
- 72
Worked answer
8
Divide the whole length by the length of one piece.
243 = 8 pieces.
Problem 7
A worker is paid $18 an hour for the first 40 hours and $27 an hour after that. What is the pay in dollars for a 46-hour week?
- $828
- $862
- $882
- $1,242
- $720
Worked answer
$882
The first 40 hours pay 40· 18 = 720 dollars.
The remaining 6 hours pay 6· 27 = 162 dollars.
The total is 720+162 = 882 dollars.
Problem 8
Two taps together fill a tank in 6 hours. One of them alone would take 10 hours. How many hours would the other take alone?
- 4
- 8
- 12
- 15
- 16
Worked answer
15
Together the taps fill 16 of the tank each hour, and the slow one fills 110.
The other fills 16-110 = 530-330 = 230 = 115 each hour.
Filling one fifteenth each hour means it takes 15 hours alone.
Problem 9
A number is doubled, then 7 is subtracted, giving 23. What was the number?
- 8
- 12
- 15
- 16
- 30
Worked answer
15
Work backwards. Before subtracting 7 the value was 23+7 = 30.
That was the number doubled, so the number itself is 302 = 15.
Check: 15 doubled is 30, and 30-7 = 23.
Problem 10
Tickets cost $9 each. A group pays $135 in total. How many tickets did they buy?
- 12
- 13
- 14
- 15
- 16
Worked answer
15
Divide the total paid by the price of one ticket.
1359 = 15 tickets.
Check: 15· 9 = 135.