Maths

Rounding and estimating

Some exam questions may ask you to give the answer in a simplified form. Rounding and estimating are two ways to make numbers easier to manage.

Rounding numbers

Giving the complete number for something is sometimes unnecessary. For instance, the attendance at a football match might be 23745. But for most people who want to know the attendance figure, an answer of 'nearly 24000', or 'roughly 23700', is fine.

football

football

We can round off large numbers like these to the nearest thousand, nearest hundred, nearest ten, nearest whole number, or any other specified number.

Round 23745 to the nearest thousand.

First, look at the digit in the thousands place. It is 3. This means the number lies between 23000 and 24000. Look at the digit to the right of the 3. It is 7. That means 23745 is closer to 24000 than 23000.

Remember

The rule is, if the next digit is: 5 or more, we 'round up'. 4 or less, it stays as it is.

23745 to the nearest thousand = 24000.

23745 to the nearest hundred = 23700.

Question

What is 23745 rounded to the nearest ten?

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Answer

Did you get the answer 23750?

If so, well done! You saw that 23745 lies between 23740 and 23750, but is closer to 23750.

If you did not get the correct answer, remember that the tens digit is 4. This means that the number lies between 23740 and 23750. The next digit is 5, so we round up: 23745 = 23750 to the nearest ten.

Decimal places

Sometimes, rather than rounding off to the nearest whole number, you might need to be a little more accurate. You might need to include some of the digits after the decimal point.

In these cases, we can round off the number up to a certain number of decimal places.

Do not confuse this with rounding off using significant figures, as this is slightly different!

Remember

The same rules for rounding up apply here:

  • 5 or more, we 'round up'.
  • 4 or less, it stays as it is.
Question

Write 2.6470588 to 2 decimal places (dp).

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Answer

Did you get the answer 2.65? You needed to round up. We want 2 decimal places.

Look at the 2nd decimal digit.

The 2nd decimal digit is 4. So the number lies between 2.64 and 2.65

Look at the next digit.

The next digit is 7, so we have to round up. So the answer is 2.65 (2 dp).

Question

On a calculator, work out So 3 over 5 = 6 over 10, giving your answer correct to one decimal place.

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Answer

On a calculator, work out So 3 over 5 = 6 over 10 = 7.874007874...

We need one decimal place. That means one number after the decimal point. The 1st number after the decimal point is 8. This means the answer lies between 7.8 and 7.9. The next digit is 7. This means we have to round up.

So the answer is So 3 over 5 = 6 over 10 = 7.9 to 1 dp.

Question

Round off the number 3.9762645 to 1 dp.

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Answer

The number lies between 3.9 and 4.0. The 7 after the 9 means you have to round up. So the answer is 4.0

Significant figures

Sometimes we do not always need to give detailed answers to problems - we just want a rough idea. When we are faced with a long number, we could round it off to the nearest thousand, or nearest million. And when we get a long decimal answer on a calculator, we could round it off to a certain number of decimal places.

Another method of giving an approximated answer is to round off using significant figures.

The word significant means: having meaning.

With the number 368249, the 3 is the most significant digit, because it tells us that the number is 3 hundred thousand and something. It follows that the 6 is the next most significant, and so on.

With the number 0.0000058763, the 5 is the most significant digit, because it tells us that the number is 5 millionths and something. The 8 is the next most significant, and so on.

Be careful however with numbers such as 30245, the 3 is the first significant figure and 0 the second, because of its value as a place holder.

We round off a number using a certain number of significant figures. The most common are 1, 2 or 3 significant figures.

Remember the rules for rounding up are the same as before:

  • If the next number is 5 or more, we round up.
  • If the next number is 4 or less, we do not round up.
Question

What would you get if you wrote the number 368249 correct to 1 significant figure?

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Answer

Did you get the answer 400000?

3 is the first significant figure, and the digit after it is more than 5, so you round up.

Question

What would you get if you wrote the number 0.00245 correct to 1 significant figure?

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Answer

Did you get the answer 0.002?

2 is the first significant figure and the digit after this is less than 5, so you do not round up.

Higher only

Question

What would you get if you wrote 0.0000058763 correct to 2 significant figures?

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Answer

Did you get the answer 0.0000059?

You had to round up the 8 to 9.

If you had problems, remember that the 2 most significant figures are 5 and 8. The digit after 8 is 7, so we have to round up 8 to 9.

So 0.0000058763 = 0.0000059 to 2 significant figures.

Question

What is 0.000030456 to two significant figures?

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Answer

Did you get the answer 0.000030?

You had to round down.

If you had problems, remember that the 2 first significant figures are 3 and 0. The digit after 0 is 4, so we have to round down.

So 0.000030456 = 0.000030 to 2 significant figures.

Question

What is 7.994 to two significant figures?

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Answer

Did you get the answer 8.0?

You had to round up.

The 2 first significant figures are 7 and 9. The digit after 9 is 9 again, so we have to round up, 7.99 rounds up to 8.00.

So 7.994 = 8.0 to 2 significant figures.

Estimating

We can use significant figures to get an approximate answer to a problem.

We can round off all the numbers in a maths problem to 1 significant figure to make 'easier' numbers. It is often possible to do this in your head.

Question

Find a rough answer to 19.4 over 0.0437

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Answer

We first round off both numbers to 1 significant figure (s.f.):

19.4 = 20 (1 s.f.)

0.0437 = 0.04 (1 s.f.)

So we now need to make the denominator a whole number. We can do this by multiplying both 20 and 0.04 by 100. 20 over 0.04 = 20 x 100 over 0.04 x 100 = 2000 over 4

Divide everything by 4.

= 2000 over 4 = 500

The real answer to 19.4 ÷ 0.0437 = 443.9359... So this was a good estimate.

Try this one. Remember, the working you do is just as important as the answer.

Question

How would you get an approximate answer for 386062 × 0.007243?

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Answer

Did you get the answer 400000 × 0.007 = 2800?

If so, well done! You rounded off correctly and worked out the approximate answer.

Rounding to 1 s.f.

386062 = 400000

0.007243 = 0.007

So 400000 × 0.007 = 2800

Upper and Lower Bounds

If we are told that a piece of wood is 12cm long to the nearest cm, then what is the range of possible lengths it could be?

It must be at least 11.5cm long to round up to 12cm. But it must be less than 12.5cm, to round down to 12cm. If it was 12.5cm or more we would round up to 13cm.

Since 12.5cm is the upper limit we call this the 'upper bound'.

So we say that 11.5cm is the lower bound and 12.5cm is the upper bound for the length of the piece of wood.

Try these questions.

Question

A fence is 22m long to the nearest metre. What are the upper and lower bounds?

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Answer

The lower bound is 21.5m and the upper bound is 22.5m

Question

A box is 8.5cm wide measured to the nearest tenth of a cm. What are the upper and lower bounds?

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Answer

Anything above 8.45cm will round up to 8.5 to the nearest tenth of a cm.

Anything less than 8.55cm will round down to 8.5cm

The lower bound is 8.45cm and the upper bound is 8.55cm

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