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.
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 at it is.
23745 to the nearest thousand = 24000.
23745 to the nearest hundred = 23700.
- Question
What is 23745 rounded to the nearest ten?
- 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 (intermediate), 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).
- 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
, giving your answer correct to one decimal place.
- Answer
On a calculator, work out
= 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
= 7.9 to 1 dp.
- Question
Round off the number 3.9762645 to 1 dp.
- 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 important. The closer a digit is to the beginning of a number, the more important - or significant - it is.
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.
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?
- Answer
Did you get the answer 400000?
3 is the first significant figure, and the digit after it is a 5, so you round up.
- Question
What would you get if you wrote the number 0.00245 correct to 1 significant figure?
- 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?
- 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.
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
- 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.
Divide everything by 4.
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?
- 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
Now try a Test Bite:Foundation or Higher