Section A
Powers of 10
Method: Follow the pattern 10³ = 1000, 10² = 100, 10¹ = 10. Each step down divides by 10, and the pattern keeps going below 10¹ into 10⁰ and then the negative powers. To find a missing power, compare the start and end numbers: how many places did the digits move, and did the number get bigger or smaller?
Example: 2.5 × 10□ = 0.025 → the digits moved 2 places smaller, which is dividing by 100, the same as multiplying by 10⁻² → the missing power is −2.
Q1
Work out 23 ÷ 10⁰
1 mark
Q2
Work out 6150 ÷ 10⁵
1 mark
Q3
Find the missing power: 0.34 × 10□ = 340
2 marks
Q4
Find the missing power: 56 × 10□ = 0.056
2 marks
Q5
Find the missing power: 9 ÷ 10□ = 900
2 marks
Q6
Which power of 10 is equal to 0.0001?
1 mark
Q7
Without working them out: is 48 × 10⁻⁵ larger or smaller than 48 × 10⁻²? Write one word.
1 mark
Q8
Three of these calculations give the same answer. Which one is different?
1 mark
POWERS OF 10 — SECTION SCORE
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Section B
Powers in Compound Percentages
Method: When the same percentage change happens every year, you multiply by the same multiplier each year. Instead of writing it out, use a power: start value × multiplierⁿ, where n is how many times the change happens. To find after how many years a value passes a target, keep multiplying by the multiplier one year at a time, counting as you go, until you cross it.
Example: $800 rises by 10% a year. After year 1: $880, year 2: $968, year 3: $1064.80 → it first goes above $1000 after 3 years.
Q9
A house is worth $150 000 and its value rises by 3% each year. What does 150 000 × 1.03⁷ represent?
1 mark
Q10
An investor uses 4000 × 1.025¹² to work out their bank balance. Interest is added once a year. For how many years has the money been invested?
1 mark
years
Q11
In the same calculation, 4000 × 1.025¹², what is the yearly interest rate, as a percentage?
1 mark
%
Q12
A town has 12 000 people. Its population is predicted to fall by 10% each year. After how many years does the population first fall below 8000?
2 marks
years
Q13
$2500 is invested at 7% interest per year, compounded annually. After how many years does the investment first go above $3500? You can use a calculator.
2 marks
years
Q14
A car worth $24 000 loses 15% of its value every year. Which calculation gives its value after n years?
1 mark
POWERS IN PERCENTAGES — SECTION SCORE
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Section C
Bounds of Whole Numbers
Method: Some things can only be counted in whole numbers — people, cars, marbles. If the exact value must be a whole number, the upper bound is the largest whole number that still rounds to the given value, not the halfway point. The lower bound is the smallest whole number that rounds to it.
Example: A whole number rounds to 50 (nearest 10). It could be 45, 46, 47, …, 54 → lower bound 45, upper bound 54. It cannot be 55, because 55 rounds up to 60.
Q15
A whole number is rounded to the nearest 10. The answer is 70. What is the upper bound?
1 mark
Q16
A whole number is rounded to the nearest 100, giving 600. How many different whole numbers could it have been?
1 mark
Q17
A newspaper reports that 3200 people attended a concert, correct to the nearest 100. What is the largest possible number of people who attended?
2 marks
people
Q18
A school has 860 students, correct to the nearest 10. What are the smallest and largest possible numbers of students?
1 mark
Q19
The crowd at a football match is 47 000, correct to the nearest 1000. What is the lower bound of the crowd?
1 mark
people
WHOLE-NUMBER BOUNDS — SECTION SCORE
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Section D
Bounds of Numbers with One Decimal Place
Method: Sometimes you are told the exact number has one decimal place. Then it can only go up in steps of 0.1, so the upper bound is the largest one-decimal-place number that still rounds to the given value.
Example: A number with one decimal place rounds to 5 (nearest whole number). It could be 4.5, 4.6, …, 5.4 → lower bound 4.5, upper bound 5.4, so 4.5 ≤ x ≤ 5.4.
Compare: if it were a measurement that rounds to 5, it could have any number of decimal places (5.49, 5.499, …), so the range would be 4.5 ≤ x < 5.5 with upper bound 5.5 — see Section E. Always check which kind of number the question says.
Q20
A number with one decimal place is rounded to the nearest whole number. The answer is 9. What is the lower bound?
1 mark
Q21
Using the same number (one decimal place, rounds to 9), what is the upper bound?
1 mark
Q22
Priya writes down a number with one decimal place. Rounded to the nearest whole number, it is 40. How many different numbers could Priya have written?
2 marks
Q23
A number with two decimal places is rounded to 1 decimal place, giving 3.6. What is its upper bound?
1 mark
ONE-DECIMAL-PLACE BOUNDS — SECTION SCORE
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Section E
Measurements & Mixed Bounds
Method: A measurement such as a length, mass or time can take any value, not just whole numbers or one decimal place. Its bounds are exactly half a unit of rounding either side. The upper bound itself would round up, so it is not included — that is why the range uses < on the right: lower bound ≤ x < upper bound. Before you find any bounds, ask: what kind of number was rounded?
Example: A shelf is 30 cm long, correct to the nearest cm → 29.5 ≤ x < 30.5.
Q24
A rope is 25 m long, correct to the nearest metre. What is the upper bound of its length?
1 mark
m
Q25
A decimal number, which could have any number of decimal places, is rounded to the nearest whole number, giving 6. Which inequality shows the range of values it could have?
1 mark
Q26
A bag of flour has a mass of 3.8 kg, correct to 1 decimal place. Which inequality shows the range of possible masses?
1 mark
Q27
Four quantities were each rounded to the nearest 10, and each gave 150. For which one is the upper bound 154?
2 marks
Q28
Arun counts the passengers on a bus: 40, to the nearest 10. Zara measures the length of the bus: 12 m, to the nearest metre. Whose upper bound could actually be the exact value? Write one name.
1 mark
MEASUREMENTS & MIXED — SECTION SCORE
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Section F · Real Life
Will the Bus Fit?
📰 The bridge that opens trucks like tin cans
In Durham, North Carolina (USA), a railway bridge crosses a street with a gap underneath of just 11 feet 8 inches — about 3.56 m. Again and again, drivers of trucks and vans that were too tall drove into it, and the bridge peeled their roofs open. It became famous online as “the can opener”. In 2019 the bridge was raised by about 20 cm, and it has still been hit since.
Height signs are rounded, and so is the height a driver knows for their bus. So how can a driver be sure they will fit? Use the simulator: set the exact heights with the sliders (they must match the rounded signs), then press Drive. Challenge: one scenario can end in a crash and the other never can. Which is which — and why?
Pick a scenario, set the exact heights, then press Drive
Q29
Scenario A. The bridge sign says 3.5 m, correct to 1 decimal place. What is the lowest the gap under the bridge could really be?
1 mark
m
Q30
Scenario A. The bus is 3.4 m tall, correct to 1 decimal place. What is the upper bound of the bus’s height?
1 mark
m
Q31
Scenario B. An old bridge sign says 4 m, correct to the nearest metre. The bus is 3.8 m tall, correct to 1 decimal place. Can the driver be certain the bus will fit?
2 marks
WILL THE BUS FIT? — SECTION SCORE
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