Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: what is |−9|?
Answer
9. The absolute value is the distance from 0, and −9 is 9 away from 0. - From Year 7: write 0.035 as a percentage.
Answer
3.5%. Multiply by 100. - From Lesson 2: round 0.04167 to 2 significant figures.
Answer
0.042. Keep 4 and 1; the next digit is 6, so round up.
Words used in this lesson
- absolute value
- the distance of a number from 0, written |x|. For example: |−7| = 7.
- distance between
- how far apart two numbers are, |a − b|. For example: |−3 − 4| = 7.
- absolute error
- the distance between a measured value and the true value. For example: |9.7 − 10| = 0.3 s.
- relative error
- the absolute error divided by the true value. For example: 0.3 ÷ 10 = 0.03.
- percentage error
- the relative error written as a percentage. For example: 0.03 × 100 = 3%.
- true value
- the correct value of a quantity. For example: the pool is 50 m.
- measured value
- the value a person or instrument records. For example: 49.6 m.
Two loaves, both 20 g short
A bakery in Accra sells 800 g loaves and 400 g rolls. A customer's kitchen scale shows 780 g for a loaf and 380 g for a roll. Both are 20 g short.
Predict first: is one shortfall worse than the other? How could you measure “worse” fairly? The first figure looks at absolute value itself: the distance from zero.
a = 3 and b = 2 point the same way, so |a + b| = |5| = 5, the same as |a| + |b| = 3 + 2 = 5.
Show me the steps
- a = 3 and b = 2 point the same way: the walk ends at 5, so |a + b| = 5, the same as |a| + |b| = 3 + 2 = 5.
- Set b = −5: the second arrow turns back. The walk ends at −2, so |a + b| = 2, less than |a| + |b| = 3 + 5 = 8.
- Set a = −4 and b = −1: both point left. The walk ends at −5, so |a + b| = 5 = |a| + |b|. Same signs give equality.
- Set b = 0: there is no second walk, so |a + b| = |a| and the two sides are equal.
|x| is the distance of x from 0, so it is never negative, and |a − b| is the distance between a and b. Two properties follow: |a × b| = |a| × |b|, and |a + b| ≤ |a| + |b|, with equality only when a and b do not have opposite signs. Solving |x − c| = d means finding the numbers d away from c: x = c + d or x = c − d.
Errors. The absolute error is |measured value − true value|. The relative error is the absolute error ÷ the true value, and the percentage error is the relative error × 100%. Always divide by the true value.
True 40, measured 42: the absolute error is |42 − 40| = 2, the relative error is 2 ÷ 40 = 0.05 and the percentage error is 5%.
Show me the steps
- True 40, measured 42: the absolute error is |42 − 40| = 2, the relative error is 2 ÷ 40 = 0.05, and the percentage error is 5%.
- Measured 38: the error is |38 − 40| = 2 again. The absolute value ignores the direction, so it is still a 5% error.
- True 20, measured 22: the same absolute error, 2, but now 2 ÷ 20 = 0.1, a 10% error. The same slip matters more on a smaller quantity.
- True 60, measured 60: no error at all, 0%.
Two common mistakes. Treating |a − b| as |a| − |b|: |2 − 7| = 5, but |2| − |7| = −5, and an absolute value cannot be negative. And dividing by the measured value instead of the true value when finding a relative or percentage error.
Worked examples
1. A recipe in Hanoi needs 250 g of rice. The kitchen scale shows 240 g. Find the absolute, relative and percentage errors.
- Absolute error: |240 − 250| = |−10| = 10 g.
- Relative error: 10 ÷ 250 = 0.04. Divide by the true value, 250.
- Percentage error: 0.04 × 100 = 4%.
2. Solve |x + 1| = 4.
- |x + 1| = |x − (−1)|, the distance from x to −1.
- The numbers 4 away from −1 are −1 + 4 = 3 and −1 − 4 = −5.
Your go. The first step is given. A plank is really 2.40 m long, and a builder measures it as 2.46 m. Find the percentage error.
- The absolute error is |2.46 − 2.40| = 0.06 m.
- Now divide by the true value and change it to a percentage.
Check the answer
0.06 ÷ 2.40 = 0.025, so the percentage error is 2.5%.
With a GDC (optional)
Most GDCs have an absolute value function, often written abs( and found in a maths or numeric menu. Typing abs(2 − 9) gives 7.
For a whole class's measurements, type the true values in one list and the measured values in a second list, then define a third list as abs(second − first) ÷ first × 100. The GDC gives every percentage error at once, and you can sort them to see whose measurement was closest.
Check before you go on
1. Which statement is true for all real numbers a and b?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Work out |3 − 11| − |−4|.
3. A student measures an eraser that is really 5.0 cm long as 4.8 cm. What is the absolute error, in cm?
Level 2 · Building
4. A taxi meter in Istanbul shows 17.2 km for a trip that is really 16 km. What is the percentage error? Give the number, without the % sign.
5. How far apart are −3.5 and 4.5 on the number line? That is, what is |−3.5 − 4.5|?
6. Which values of x solve |x − 2| = 5?
Level 3 · Stretch
7. A runner's watch measures a 400 m lap of a track in Nairobi with a percentage error of 2%, and it reads more than the true distance. What does the watch read, in metres?
8. Two surveyors in Kuala Lumpur measure a 50 m swimming pool as 49.6 m and 50.3 m. What is the smaller of their two percentage errors? Give the number, without the % sign.
Something to think about
Can a relative error be bigger than 1? What would that say about the measurement?
One way to think about it
Yes. If a length is really 5 cm and someone records 12 cm, the absolute error is |12 − 5| = 7 cm and the relative error is 7 ÷ 5 = 1.4, a 140% error. A relative error above 1 means the error is bigger than the thing being measured, which usually points to a wrong unit or a misread scale rather than a careless reading.
Exit ticket
9. A market scale in Lima reads 2.1 kg for a bag of rice whose true mass is 2.5 kg. What is the percentage error?