For a product of positive rounded measurements, the lower bound multiplies the two lower bounds and the upper bound multiplies the two upper bounds. This is how you find the smallest and largest possible area of a rectangle, cost from a price and a mass, or distance from speed and time.
It follows from bounding a sum and a difference, where pairing mattered more, and it is the core calculation in precision bounds and measurement.
Why does the product use like with like?
Take positive numbers only. Making either factor bigger makes the product bigger, so the largest product needs both factors at their largest.
- lower bound of a × b = aL × bL
- upper bound of a × b = aU × bU
For a quotient, the bottom value works the other way round. To make a ÷ b small, divide by a large b.
- lower bound of a ÷ b = aL ÷ bU
- upper bound of a ÷ b = aU ÷ bL
Worked example
A rectangle measures 6.4 m by 3.2 m, each to 1 decimal place. Find the bounds of its area and say what accuracy the area can be given to.
Step 1, intervals: 6.35 ≤ length < 6.45 and 3.15 ≤ width < 3.25.
Step 2, lower bound: 6.35 × 3.15 = 20.0025.
Step 3, upper bound: 6.45 × 3.25 = 20.9625.
Step 4, interval: 20.0025 ≤ area < 20.9625 m².
Step 5, accuracy: to 2 significant figures the bounds are 20 and 21, which differ. To 1 significant figure both are 20. So the area is 20 m² to 1 significant figure, and the calculator’s 20.48 claims more than the data supports.
The mistake to watch for
A common slip is mixing a lower bound from one measurement with an upper bound from the other.
Mistaken working: lower bound = 6.35 × 3.25 = 20.6375.
The student used the smallest length with the largest width. That product is not the smallest possible area, because the smallest width gives 20.0025.
Always keep the pairing consistent for a product: both lower, or both upper. Write “L × L” and “U × U” beside each line so you can check at a glance.
Do not round the intervals before multiplying. Keep 6.35 and 3.15, not 6.3 and 3.2.
Check yourself
Try these, then open each answer.
1. A rectangle is 12 cm by 5 cm, each to the nearest centimetre. Find the bounds of its area.
Show answer
Intervals: 11.5 ≤ length < 12.5 and 4.5 ≤ width < 5.5. Lower bound 11.5 × 4.5 = 51.75. Upper bound 12.5 × 5.5 = 68.75.
51.75 ≤ area < 68.75 cm²
2. Apples cost RM4.8 per kg and the mass is 2.5 kg, each to 1 decimal place. Find the bounds of the total cost.
Show answer
Intervals: 4.75 ≤ price < 4.85 and 2.45 ≤ mass < 2.55. Lower bound 4.75 × 2.45 = 11.6375. Upper bound 4.85 × 2.55 = 12.3675.
RM11.6375 ≤ cost < RM12.3675, which is about RM11.64 to RM12.37.
3. A car travels 150 m (nearest 10 m) in 20 s (nearest second). Find the bounds of its speed, to 3 significant figures.
Show answer
Intervals: 145 ≤ distance < 155 and 19.5 ≤ time < 20.5. Lower bound 145 ÷ 20.5 = 7.073… so 7.07. Upper bound 155 ÷ 19.5 = 7.948… so 7.95.
7.07 ≤ speed < 7.95 m/s
Where this leads next
The last skill in the set is about what the calculator display does and does not tell you: distinguish accuracy from displayed decimal places. The bounds and rounding explainer shows the product and quotient pairings side by side.
Some students learn this rule well but still apply a sum pairing to a product when tired. A teacher in online one-to-one Mathematics tuition can spot that switch in your written working and help you catch it yourself.