Bounds Patrol

Bounds Patrol

Real measurements are never perfectly exact, so we round them, estimate with them, and state the interval they truly lie in. The last of those three is the one that decides real questions. This module ends by asking whether a box measured at 90 cm will fit on a shelf measured at 90 cm, and the answer is not the one most people give.

Decimal places and significant figures

Two ways to round, and the difference shows up most clearly on the same number. Take 0.02307:

Round to 2 decimal places

Round 3.14159 to 2 decimal places.

Round to 2 significant figures

Round 0.00408 to 2 significant figures.

Significant figures

Round 34,700 to 2 significant figures.

• 35,000
• 34,000
• 35
• 34,700

Estimating

To estimate a calculation quickly, round every number to 1 significant figure and then work it out. Write the result with the ≈ sign, meaning approximately equal. The point is not to replace the real calculation. It is to have a rough answer in your head before you press the buttons, so that a mistyped digit or a misplaced decimal point announces itself instead of being written down. It works on divisions as well as products, and choosing convenient roundings is allowed: rounding a divisor to something that divides neatly is sensible, not cheating.

Estimate the product

Estimate 31 × 19 by rounding each number to 1 significant figure. What is your estimate?

Estimate the division

Estimate 4128 ÷ 19 by rounding each number to 1 significant figure. What is your estimate?

Catching the slip

A student needs 4128 ÷ 19. They type it into a calculator and write down 21.7. The estimate you just made says the answer should be somewhere near 200. Twenty-one is not near two hundred, so something went wrong, and the size of the gap says what: the answer is out by a factor of about ten, which usually means a mistyped digit or a decimal point in the wrong place. They try again and get 217.3, which sits comfortably beside the estimate. The estimate did not produce the answer. It caught the wrong one, which is the job it exists to do, and it takes about four seconds.

Which answer is wrong?

A student calculates 6.2 × 48 and writes 29.76. Without doing the exact calculation, what can you say?

• It is wrong: rounding gives roughly 6 × 50 = 300, and 29.76 is about ten times too small, which suggests a decimal point in the wrong place
• It looks about right, since 29.76 is close to 30 and both numbers are smallish
• Nothing can be said without doing the exact calculation
• It is wrong, and the true answer must be around 30,000

Error intervals

A rounded measurement hides a range, and the range is always half the rounding unit either side. The four terms below are the ones exam questions use:

Match each measurement to its interval

Why not ≤ at both ends?

The error interval for 12 cm measured to the nearest cm is written 11.5 ≤ x < 12.5. Why is the lower bound included but the upper bound excluded?

• A true length of exactly 12.5 cm would round UP to 13, not down to 12, so it could never have been recorded as 12. A length of exactly 11.5 cm does round to 12, so it belongs in the interval
• It is an arbitrary convention, adopted so that intervals do not overlap
• Because no measuring instrument can produce a value of exactly 12.5
• To be cautious, since the upper end is riskier in real problems

Will it fit?

A box has to go on a shelf. Everything here turns on bounds rather than on the numbers written down.

• The shelf is 90 cm long and the box is 90 cm long, both measured to the nearest cm. Will the box definitely fit?
• What would have to be true for the fit to be guaranteed?
• Both are re-measured to the nearest millimetre: the shelf is 900 mm and the box is 895 mm. Does it fit now?
• Why does a question like this have to be settled on worst cases rather than the stated measurements?

Which are true?

Select ALL THREE statements that are TRUE.

• The error interval always extends half the rounding unit either side, whatever the unit is
• An estimate is worth making even when you have a calculator, because it catches wrong answers rather than producing right ones
• To guarantee that one measured object fits inside another, you must compare the worst cases rather than the stated values
• Leading zeros count as significant figures, since they are written down
• Both bounds of an error interval are included, so 12.5 is a possible length for something recorded as 12 cm
• Rounding 34,700 to 2 significant figures makes it a much smaller number

The accuracy rules

To estimate a calculation, round each number to 1 _____ figure first, and use the result to check that a calculator answer is sensible. When counting significant figures, leading _____ do not count. A length given as 12 cm to the nearest cm has an error interval of 11.5 ≤ x < 12.5: the lower bound is included, while the upper bound is _____ because 12.5 would round up to 13. To be sure one object fits inside another, compare the _____ cases rather than the stated measurements.