Problem Solving — Practice
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 questions 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 number is doubled, then 5 is added, giving 23. What was the number?
Worked answer
9
23 - 5 = 18
18 ÷ 2 = 9
Check: 9 × 2 + 5 = 23 ✓
Problem 2
What comes next in 1, 4, 9, 16, …?
Worked answer
25
12, 22, 32, 42, …
Next is 52 = 25
Problem 3
You have 3 shirts and 4 pairs of trousers. How many outfits?
Worked answer
12
3 × 4 = 12
Problem 4
A number is halved, then 3 is subtracted, giving 7. What was the number?
Worked answer
20
7 + 3 = 10
10 × 2 = 20
Problem 5
What comes next in 2, 5, 8, 11, …?
Worked answer
14
It grows by 3
11 + 3 = 14
Problem 6
Nina spent half her money on a game, then $7 on lunch, and has $18 left. How much did she start with?
Worked answer
50
Undo lunch: 18 + 7 = 25
Undo the half: 25 × 2 = 50
Check: half of 50 is 25, less 7 is 18 ✓
Problem 7
At a party, each of the 12 guests shakes hands with every other guest exactly once. How many handshakes?
Worked answer
66
Each of 12 shakes 11 hands: 132
Every handshake got counted twice
132 ÷ 2 = 66
Problem 8
Adult tickets cost $8, child tickets $5. 20 tickets sold for $136. How many adults?
Worked answer
12
All 20 as children: 100. We need 136, so 36 more
Each child swapped for an adult adds 8 - 5 = 3
36 ÷ 3 = 12 adults
Problem 9
Leo spent 13 of his money, then $12 more, and has $20 left. How much did he start with?
Worked answer
48
20 + 12 = 32 before the $12
32 is 23 of the start
32 ÷ 23 = 48
Problem 10
Figure n of a pattern has 2n + 1 dots. How many dots in figure 25?
Worked answer
51
2(25) + 1 = 51
Problem 11
Two consecutive whole numbers add to 35. What is the smaller one?
Worked answer
17
n + (n+1) = 35
2n = 34, so n = 17
The numbers are 17 and 18.
Problem 12
How many diagonals does a hexagon have?
Worked answer
9
Each vertex connects to 6 - 3 = 3 others by a diagonal
6 × 3 = 18, counting each diagonal twice
18 ÷ 2 = 9
Problem 13
A frog at the bottom of a 10 ft well climbs 3 ft each day and slips back 2 ft each night. On which day does it get out?
Worked answer
8
Net gain is 1 ft per full day
After 7 days it starts the day at 7 ft
On day 8 it climbs 3 ft to 10 ft and is out — before slipping
So 8 days, not 10.
Problem 14
A number is multiplied by 3, then 4 is subtracted, then the result is halved, giving 10. Find the number.
Worked answer
8
Undo the halving: 10 × 2 = 20
Undo the subtraction: 20 + 4 = 24
Undo the multiplication: 24 ÷ 3 = 8
Problem 15
A jar holds 25 coins, only dimes and quarters, worth $4.30. How many quarters?
Worked answer
12
All 25 as dimes: 250 cents
We need 430, which is 180 more
Each swap adds 25 - 10 = 15 cents
180 ÷ 15 = 12 quarters
Problem 16
Find the sum 1 + 2 + 3 + ⋯ + 50.
Worked answer
1275
Pair them: 1+50 = 51, 2+49 = 51, …
There are 25 such pairs
25 × 51 = 1275