Chapter 8 of 8

All 8 chapters
  1. 1Laws of powers, readunderstood
  2. 2Zero and negative powers, readunderstood
  3. 3Roots and fractional powers, readunderstood
  4. 4Standard form, readunderstood
  5. 5Calculating in standard form, readunderstood
  6. 6Rational and irrational numbers, readunderstood
  7. 7Surds, readunderstood
  8. 8Accuracy and bounds, readunderstood

M4.1.8 Chapter 8

Accuracy and bounds

A rod is 7.4 cm long to 1 decimal place. How long could it really be?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Round a number to 2 or 3 significant figures
  • Find the bounds of 7.4 cm rounded to 1 decimal place
  • Find the largest possible area from rounded lengths

Sipho measures a rod and writes 7.4 cm, correct to 1 decimal place. He asks if the rod is exactly 7.4 cm. It almost never is. First, how rounding to significant figures works.

The significant figures of a number are its digits, counted from the first digit that is not zero. In 0.04567 the first significant figure is 4. We write s.f. for significant figures and d.p. for decimal places.

Worked example

Round 0.04567 to 2 significant figures.

  1. The first digit that is not zero is 4. It is the first significant figure.
  2. Count 2 significant figures: 4 and 5. The next digit is 6.
  3. The 6 is 5 or more, so round up: 0.046.

Answer: 0.046

A table with the head Number, 3 s.f. and 2 s.f. Rows: 3 478, 3 480, 3 500; 0.04567, 0.0457, 0.046; 62 150, 62 200, 62 000.Number3 s.f.2 s.f.3 4783 4803 5000.045670.04570.04662 15062 20062 000
The zeros after a rounded big number hold the places. They are not significant figures.

Now back to the rod. Rounding to 1 d.p. means the numbers are measured in steps of 0.1 cm. A rod of 7.4 cm could really be anywhere that rounds to 7.4.

Draw it first

Find the bounds of 7.4 cm

  1. Step 1: the label 7.4 cm to 1 decimal place. Step 2: the label half a step is 0.05 cm. Step 3: a number line with ticks at 7.3 cm, 7.4 cm and 7.5 cm, a black dot at 7.35 cm marked lower bound and a ring at 7.45 cm marked upper bound. Step 4: the label 7.35 cm ≤ length < 7.45 cm.7.4 cm to 1 decimal place

    1 The rod is 7.4 cm to 1 decimal place. The step between rounded numbers is 0.1 cm.

  2. Step 1: the label 7.4 cm to 1 decimal place. Step 2: the label half a step is 0.05 cm. Step 3: a number line with ticks at 7.3 cm, 7.4 cm and 7.5 cm, a black dot at 7.35 cm marked lower bound and a ring at 7.45 cm marked upper bound. Step 4: the label 7.35 cm ≤ length < 7.45 cm.7.4 cm to 1 decimal placehalf a step is 0.05 cm

    2 Half a step is 0.05 cm. The true length is within 0.05 cm of 7.4 cm.

  3. Step 1: the label 7.4 cm to 1 decimal place. Step 2: the label half a step is 0.05 cm. Step 3: a number line with ticks at 7.3 cm, 7.4 cm and 7.5 cm, a black dot at 7.35 cm marked lower bound and a ring at 7.45 cm marked upper bound. Step 4: the label 7.35 cm ≤ length < 7.45 cm.7.4 cm to 1 decimal placehalf a step is 0.05 cm7.3 cm7.4 cm7.5 cm7.35 cm7.45 cmlower boundupper bound

    3 Mark 7.35 cm and 7.45 cm. The black dot is included and the ring is not.

  4. Step 1: the label 7.4 cm to 1 decimal place. Step 2: the label half a step is 0.05 cm. Step 3: a number line with ticks at 7.3 cm, 7.4 cm and 7.5 cm, a black dot at 7.35 cm marked lower bound and a ring at 7.45 cm marked upper bound. Step 4: the label 7.35 cm ≤ length < 7.45 cm.7.4 cm to 1 decimal placehalf a step is 0.05 cm7.3 cm7.4 cm7.5 cm7.35 cm7.45 cmlower boundupper bound7.35 cm ≤ length < 7.45 cm

    4 So the length is at least 7.35 cm and less than 7.45 cm.

Why is the upper end left out? A rod of exactly 7.45 cm would round up to 7.5 cm. And 7.35 cm rounds up to 7.4 cm, so the lower end is included. That is why we write ≤ at the bottom and < at the top.

Worked example

Find the bounds of 7.4 cm rounded to 1 decimal place.

  1. One decimal place means the step is 0.1 cm.
  2. Half a step is 0.05 cm.
  3. The lower bound is 7.4 − 0.05 = 7.35 cm.
  4. The upper bound is 7.4 + 0.05 = 7.45 cm.
  5. So 7.35 cm ≤ length < 7.45 cm.

Answer: 7.35 cm ≤ length < 7.45 cm

The half step depends on how the number was rounded. Here are some common ones.

See it three ways

With things

Look at a ruler marked every 0.1 cm. A true length is rounded to the nearest mark, so it can be half a mark away.

As a picture

A table with the head Rounded to and Half step. Rows: 1 d.p., 0.05; 2 d.p., 0.005; whole number, 0.5; nearest 10, 5.Rounded toHalf step1 d.p.0.052 d.p.0.005whole number0.5nearest 105

In symbols

7.4 rounded to 1 d.p.: 7.4 − 0.05 = 7.35 and 7.4 + 0.05 = 7.45.

A number line with ticks at 240 g, 250 g and 260 g, a black dot at 245 g marked lower bound and a ring at 255 g marked upper bound.240 g250 g260 g245 g255 glower boundupper bound
250 g to the nearest 10 g: at least 245 g, and less than 255 g.

Worked example

A bag weighs 250 g, to the nearest 10 g. Find the bounds of its mass.

  1. The step is 10 g, so half a step is 5 g.
  2. The lower bound is 250 − 5 = 245 g.
  3. The upper bound is 250 + 5 = 255 g.
  4. So 245 g ≤ mass < 255 g.

Answer: 245 g ≤ mass < 255 g

Now an area. A rectangle is 6 cm by 4 cm, each side to the nearest cm. Each side can be half a centimetre bigger or smaller.

A rectangle drawn at its upper bounds, 6.5 cm wide and 4.5 cm high, with a hatched ring. A white rectangle inside, from the same top left corner, is at the lower bounds, 5.5 cm wide and 3.5 cm high. The four sides are labelled 6.5 cm, 4.5 cm, 5.5 cm and 3.5 cm.6.5 cm4.5 cm5.5 cm3.5 cm
The big outline uses both upper bounds. The white rectangle uses both lower bounds. The true rectangle is in between.

Area is length times width. A bigger length and a bigger width give a bigger area. So the largest area uses both upper bounds, and the smallest area uses both lower bounds.

Worked example

A rectangle measures 6 cm by 4 cm, each to the nearest cm. Find the largest possible area.

  1. Each side is to the nearest cm, so half a step is 0.5 cm.
  2. The upper bound of 6 cm is 6 + 0.5 = 6.5 cm.
  3. The upper bound of 4 cm is 4 + 0.5 = 4.5 cm.
  4. The largest area uses both upper bounds: 6.5 × 4.5 = 29.25.
  5. So the largest possible area is 29.25 cm². The true area is always just under it, so we call it the upper bound of the area.

Answer: Largest possible area: 29.25 cm²

Each side stays just below its upper bound. So the true area can be as close to 29.25 cm² as you like, but it stays under it. The lower bound of the area is 19.25 cm², from 5.5 × 3.5. The true area is at least that.

Try it

Which numbers round to 7.4?

You need: paper and a pencil.

  1. Write the four numbers 7.34, 7.35, 7.44, 7.45.
  2. Round each one to 1 decimal place. A number halfway between rounds up.
  3. Mark which ones give 7.4.

What you will see: 7.34 rounds to 7.3 and 7.35 rounds to 7.4. 7.44 rounds to 7.4 and 7.45 rounds to 7.5. The lowest that gives 7.4 is 7.35, and 7.45 is the first to give 7.5.

If you cannot try it, this is what happens.
A table with the head Number and To 1 d.p. Rows: 7.34, 7.3; 7.35, 7.4; 7.44, 7.4; 7.45, 7.5.NumberTo 1 d.p.7.347.37.357.47.447.47.457.5

A wrong idea many people have

The upper bound of 7.4 is 7.5.

Why it feels right: It seems that the next number in the steps of 0.1 should be the top of the stretch.

The test that breaks it: Take 7.46. It rounds to 7.5, not 7.4. So 7.46 cannot be a length of 7.4 cm to 1 d.p.

What is true: The stretch reaches only half a step above 7.4. The upper bound is 7.45.

A wrong idea many people have

0.004051 to 2 significant figures is 0.00.

Why it feels right: It seems that the first two digits after the point are the first two significant figures.

The test that breaks it: 0.00 is 0, and that loses all the size of the number. The zeros at the front hold places. They are not significant.

What is true: The first significant figure is 4. The first two are 4 and 0, and the next digit is 5, so round up: 0.0041.

Key words

significant figures
the digits of a number counted from the first one that is not zero.
lower bound
the smallest value a rounded number can stand for. It is included.
upper bound
the value where a rounded number stops. It is not included.

Back to your guess

The answer is C: Anything from 7.35 cm up to, but not including, 7.45 cm.

Rounding to 1 decimal place means the true length is within half a step, 0.05 cm, of 7.4 cm. So it is at least 7.35 cm and less than 7.45 cm. Written as 7.35 ≤ length < 7.45.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Round 6 284 to 2 significant figures.

    Tap your answer.

    1. Not this one. You rounded down. The digit after the second significant figure is 8, which is 5 or more, so round up. Go to M4.1.8. Try another.

    2. Not this one. You dropped the zeros. The zeros hold the places, so 6 300 stays a number in the thousands. Go to M4.1.8. Try another.

    3. Not this one. This is 3 significant figures. The question asks for 2. Go to M4.1.8. Try another.

    4. Yes, that is right. The first 2 significant figures are 6 and 2. The next digit is 8, so round up to 6 300.

    See the answer

    6 300

    The first 2 significant figures are 6 and 2. The next digit is 8, so round up to 6 300.

    • If you chose A: You rounded down. The digit after the second significant figure is 8, which is 5 or more, so round up. Go to M4.1.8.
    • If you chose B: You dropped the zeros. The zeros hold the places, so 6 300 stays a number in the thousands. Go to M4.1.8.
    • If you chose C: This is 3 significant figures. The question asks for 2. Go to M4.1.8.

  2. A length is 12.6 m, rounded to 1 decimal place. Which line shows its bounds?

    Tap your answer.

    1. Not this one. You went a whole step of 0.1 each way. The stretch reaches only half a step, 0.05, each way. Go to M4.1.8. Try another.

    2. Yes, that is right. Half a step is 0.05 m. 12.6 − 0.05 = 12.55 and 12.6 + 0.05 = 12.65.

    3. Not this one. This starts at 12.6 and runs one whole step up. Rounding goes both ways, so the stretch starts below 12.6. Go to M4.1.8. Try another.

    See the answer

    12.55 m ≤ length < 12.65 m

    Half a step is 0.05 m. 12.6 − 0.05 = 12.55 and 12.6 + 0.05 = 12.65.

    • If you chose A: You went a whole step of 0.1 each way. The stretch reaches only half a step, 0.05, each way. Go to M4.1.8.
    • If you chose C: This starts at 12.6 and runs one whole step up. Rounding goes both ways, so the stretch starts below 12.6. Go to M4.1.8.

  3. A mass is 460 g, to the nearest 10 g. Which line shows its bounds?

    Tap your answer.

    1. Yes, that is right. Half a step is 5 g. 460 − 5 = 455 and 460 + 5 = 465.

    2. Not this one. You went a whole step of 10 each way. The stretch reaches only half a step, 5, each way. Go to M4.1.8. Try another.

    3. Not this one. This only goes upwards. A mass of 457 g would also round to 460 g, so the stretch starts below 460 g. Go to M4.1.8. Try another.

    See the answer

    455 g ≤ mass < 465 g

    Half a step is 5 g. 460 − 5 = 455 and 460 + 5 = 465.

    • If you chose B: You went a whole step of 10 each way. The stretch reaches only half a step, 5, each way. Go to M4.1.8.
    • If you chose C: This only goes upwards. A mass of 457 g would also round to 460 g, so the stretch starts below 460 g. Go to M4.1.8.

  4. The picture shows a rectangle of 5 cm by 4 cm, each to the nearest cm. What is its largest possible area?

    A rectangle drawn at its upper bounds, 5.5 cm wide and 4.5 cm high, with a hatched ring. A white rectangle inside, from the same top left corner, is at the lower bounds, 4.5 cm wide and 3.5 cm high. The four sides are labelled 5.5 cm, 4.5 cm, 4.5 cm and 3.5 cm.5.5 cm4.5 cm4.5 cm3.5 cm
    Tap your answer.

    1. Not this one. This is the area of the rounded sides, 5 × 4. The largest possible area uses the upper bounds, which are bigger. Go to M4.1.8. Try another.

    2. Yes, that is right. The upper bounds are 5.5 cm and 4.5 cm. 5.5 × 4.5 = 24.75.

    3. Not this one. This is the smallest possible area, from the lower bounds. The largest uses the upper bounds. Go to M4.1.8. Try another.

    See the answer

    24.75 cm²

    The upper bounds are 5.5 cm and 4.5 cm. 5.5 × 4.5 = 24.75.

    • If you chose A: This is the area of the rounded sides, 5 × 4. The largest possible area uses the upper bounds, which are bigger. Go to M4.1.8.
    • If you chose C: This is the smallest possible area, from the lower bounds. The largest uses the upper bounds. Go to M4.1.8.

  5. Which of these could NOT be a length of 7.4 cm to 1 decimal place?

    Tap your answer.

    1. Not this one. 7.35 is exactly halfway, and a halfway number rounds up to 7.4. So it can be a length of 7.4 cm. Go to M4.1.8. Try another.

    2. Not this one. 7.449 is below 7.45, so it rounds down to 7.4. It can be a length of 7.4 cm. Go to M4.1.8. Try another.

    3. Yes, that is right. 7.45 is halfway between 7.4 and 7.5, and a halfway number rounds up. So it rounds to 7.5, not 7.4.

    See the answer

    7.45 cm

    7.45 is halfway between 7.4 and 7.5, and a halfway number rounds up. So it rounds to 7.5, not 7.4.

    • If you chose A: 7.35 is exactly halfway, and a halfway number rounds up to 7.4. So it can be a length of 7.4 cm. Go to M4.1.8.
    • If you chose B: 7.449 is below 7.45, so it rounds down to 7.4. It can be a length of 7.4 cm. Go to M4.1.8.

  6. From an earlier chapter

    Write 0.00072 in standard form. What is the power of ten?

    See the answer

    −4

    The point moves 4 places right, to after the 7. So 0.00072 = 7.2 × 10⁻⁴.

  7. Round 0.04587 to 3 significant figures.

    See the answer

    0.0459

    The first significant figure is 4. The first 3 are 4, 5 and 8, and the next digit is 7, which is 5 or more. So round up to 0.0459.

  8. A mass is 460 g, to the nearest 10 g. What is its upper bound in g?

    See the answer

    465

    Half a step is 5 g. 460 + 5 = 465. The mass is below this value, but as near to it as you like.

  9. A rectangle measures 6 cm by 4 cm, each to the nearest cm. What is its smallest possible area in cm²?

    See the answer

    19.25

    The smallest area uses both lower bounds: 5.5 cm and 3.5 cm. 5.5 × 3.5 = 19.25.

Keep it

Copy this into your exercise book.

Rounded values stand for a stretch: half a step each way. 7.4 to 1 d.p.: 7.35 ≤ length < 7.45. The largest area uses both upper bounds. The smallest uses both lower.

Draw: A number line with ticks 7.3, 7.4 and 7.5, a black dot at 7.35, and a ring at 7.45.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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