When rounding, how do you decide between the two numbers on either side?

Look at the digit immediately to the right of the place value you are rounding to. If it is 5 or more you round up to the bigger number, and if it is less than 5 you round down to the smaller one.

Round 1294 to the nearest 100.

Counting in hundreds, 1294 lies between 1200 and 1300, and the digit in the tens column is 9. That sends it up, so 1294 to the nearest 100 is 1300 .

Round 1798 to the nearest 10, and say why it needs care.

Counting in tens, 1798 lies between 1790 and 1800, and the 8 in the units column sends it up to 1800 . The care is needed because the 9 in the tens column rolls over, so the hundreds digit changes as well.

Why is 2.395 rounded to 2 decimal places written as 2.40 rather than 2.4?

An answer given to 2 decimal places must show two digits after the point, so the zero is part of the answer. It records the accuracy the answer was given to, which 2.4 on its own would not.

True or False? 1.267 rounded to 2 decimal places is 1.270.

False. The answer is 1.27 , since the rounding stops at the second decimal place. A third decimal place would claim an accuracy that the rounding does not provide.

What is the first significant figure of 0.006207?

The 6, because the first significant figure is the digit in the biggest place value that is not zero. The zeros before the 6 only hold the place and are not significant.

True or False? In 0.006207 the zero between the 2 and the 7 is a significant figure.

True. Only the zeros before the first significant figure fail to count, because those merely hold the place. The zero between the 2 and the 7 is the third significant figure of 0.006207.

Which digit is the third significant figure of 3097, and why?

The 9, because you count digits to the right starting from the first significant figure. Here 3 is the first, 0 is the second and 9 is the third.

Why is 34 568 to 2 significant figures written as 35 000?

The two significant figures are 3 and 5, and the three zeros fill the remaining place values up to the decimal point. Without them the answer would read 35, which is a completely different size.

Round 0.003435 to 3 significant figures.

The three significant figures are 3, 4 and 3, and the next digit is 5, so the last of them rounds up to give 0.00344 . The two zeros after the decimal point are kept because they fix the place value of the first significant figure.

What is an estimate useful for, apart from avoiding a hard calculation?

It gives a value to check a full answer against. If the calculated answer comes out much bigger or much smaller than the estimate, there is a mistake somewhere in the working.

In estimating a division, when can you be sure the estimate is an overestimate?

When the number being divided is rounded up and the number you are dividing by is rounded down . Rounding both of them the same way leaves the effect genuinely unclear, since the two changes pull the answer in opposite directions.

True or False? When estimating, it is fine to round a small decimal down to zero.

False. Rounding a value to zero throws away everything it contributed, so the estimate stops being worth anything. It is worst when the small decimal is a denominator , because dividing by zero is undefined.

When is it sensible to round to something other than 1 significant figure?

When another value makes the arithmetic easier, so 16.2 can be taken as 15 and 1180 as 1200. The aim is a calculation you can do in your head, not the closest possible rounding.

Define the upper bound of a rounded value.

The upper bound is the value that the true measurement must be smaller than. It is the top of the range of values that would all round to the number you were given.

Define the lower bound of a rounded value.

The lower bound is the smallest value that the true measurement could actually be. It is the bottom of the range of values that would all round to the number you were given.

A length is given as 27 m to the nearest metre. What is the degree of accuracy?

The degree of accuracy is 1 metre , since that is the size of the step the value was rounded to. A degree of accuracy can be stated as a measure like this one, or as a place value such as 2 significant figures.

How do you find the upper and lower bounds of a rounded value?

Halve the degree of accuracy the value was rounded to, then add that to the value for the upper bound and subtract it for the lower bound. A mass of 24 800 g given to the nearest 100 g has half of 100 as 50, so its bounds are 24 850 g and 24 750 g.

O LevelMathsCambridge (CIE)FlashcardsNumberRounding, Estimation & Bounds

Rounding, Estimation & Bounds (Cambridge (CIE) O Level Maths): Flashcards

Exam code: 4024

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FrontRounding to a Given Place Value
When rounding, how do you decide between the two numbers on either side?

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When rounding, how do you decide between the two numbers on either side?
Look at the digit immediately to the right of the place value you are rounding to.
If it is 5 or more you round up to the bigger number, and if it is less than 5 you round down to the smaller one.
Round 1294 to the nearest 100.
Counting in hundreds, 1294 lies between 1200 and 1300, and the digit in the tens column is 9.
That sends it up, so 1294 to the nearest 100 is 1300.
Round 1798 to the nearest 10, and say why it needs care.
Counting in tens, 1798 lies between 1790 and 1800, and the 8 in the units column sends it up to 1800.
The care is needed because the 9 in the tens column rolls over, so the hundreds digit changes as well.
Complete the working for rounding 7.82741 to 3 decimal places:
Counting in thousandths, the number lies between 7.827 and $______$ , and since the next digit is 4 the answer is $______$ to 3 decimal places.
The completed working is:
Counting in thousandths, the number lies between 7.827 and 7.828, and since the next digit is 4 the answer is 7.827 to 3 decimal places.
Why is 2.395 rounded to 2 decimal places written as 2.40 rather than 2.4?
An answer given to 2 decimal places must show two digits after the point, so the zero is part of the answer.
It records the accuracy the answer was given to, which 2.4 on its own would not.
True or False?
1.267 rounded to 2 decimal places is 1.270.
False.
The answer is 1.27, since the rounding stops at the second decimal place.
A third decimal place would claim an accuracy that the rounding does not provide.
What is the first significant figure of 0.006207?
The 6, because the first significant figure is the digit in the biggest place value that is not zero.
The zeros before the 6 only hold the place and are not significant.
True or False?
In 0.006207 the zero between the 2 and the 7 is a significant figure.
True.
Only the zeros before the first significant figure fail to count, because those merely hold the place.
The zero between the 2 and the 7 is the third significant figure of 0.006207.
Which digit is the third significant figure of 3097, and why?
The 9, because you count digits to the right starting from the first significant figure.
Here 3 is the first, 0 is the second and 9 is the third.
Why is 34 568 to 2 significant figures written as 35 000?
The two significant figures are 3 and 5, and the three zeros fill the remaining place values up to the decimal point.
Without them the answer would read 35, which is a completely different size.
Round 0.003435 to 3 significant figures.
The three significant figures are 3, 4 and 3, and the next digit is 5, so the last of them rounds up to give 0.00344.
The two zeros after the decimal point are kept because they fix the place value of the first significant figure.
Complete the rule for how accurate an answer should be:
When a question does not say what accuracy to use, give the final answer to $______$ significant figures, having kept at least $______$ significant figures throughout the working.
The completed rule is:
When a question does not say what accuracy to use, give the final answer to three significant figures, having kept at least four significant figures throughout the working.
When may an answer be left without rounding it at all?
An answer that is an exact value needs no rounding, which covers a simplified fraction such as $\frac{5}{6}$ and anything left in terms of $π$ or a square root, for example $4 π$ or $\sqrt{3}$ itself.
An exact decimal of up to five significant figures, such as 0.9375, can also be written out in full.
What accuracy is usually expected for an answer in money, and for an angle?
Money is given to 2 decimal places unless the context suggests otherwise, so £64.749214 becomes £64.75 as the answer.
Angles are given to 1 decimal place, so $43.5789 °$ becomes $43.6 °$ instead.
A class of 31 students needs at least one adult for every 10 students. Why is the answer 4 adults rather than 3?
Dividing gives $31 \div 10 = 3.1$ adults, and the context forces this up to 4, because three adults would only cover 30 students.
A context can force a round down just as easily: 50 apples packed in crates of 12 gives $50 \div 12 = 4.16 \dots$ but only 4 crates can actually be filled.
What is the general rule for estimating the value of a calculation?
Round every number to 1 significant figure and then carry out the calculation with the rounded values.
So $41.3 \div 9.79$ becomes $40 \div 10$ and the estimate is 4.
What is an estimate useful for, apart from avoiding a hard calculation?
It gives a value to check a full answer against.
If the calculated answer comes out much bigger or much smaller than the estimate, there is a mistake somewhere in the working.
Complete the estimate, rounding each number to 1 significant figure:
$\frac{17.3 \times 3.81}{11.5} \approx \frac{______\times 4}{______}$
The completed estimate is:
$\frac{17.3 \times 3.81}{11.5} \approx \frac{20 \times 4}{10} = 8$
Note that 11.5 rounds to 10 rather than to 12, because only the leading digit survives to 1 significant figure.
You estimate $18 \times 7.8$ by rounding both numbers up. Is that an overestimate or an underestimate?
An overestimate, because in a product making either factor larger makes the answer larger.
Rounding both numbers down instead would give an underestimate.
In estimating a division, when can you be sure the estimate is an overestimate?
When the number being divided is rounded up and the number you are dividing by is rounded down.
Rounding both of them the same way leaves the effect genuinely unclear, since the two changes pull the answer in opposite directions.
True or False?
When estimating, it is fine to round a small decimal down to zero.
False.
Rounding a value to zero throws away everything it contributed, so the estimate stops being worth anything.
It is worst when the small decimal is a denominator, because dividing by zero is undefined.
When is it sensible to round to something other than 1 significant figure?
When another value makes the arithmetic easier, so 16.2 can be taken as 15 and 1180 as 1200.
The aim is a calculation you can do in your head, not the closest possible rounding.
Define the upper bound of a rounded value.
The upper bound is the value that the true measurement must be smaller than.
It is the top of the range of values that would all round to the number you were given.
Define the lower bound of a rounded value.
The lower bound is the smallest value that the true measurement could actually be.
It is the bottom of the range of values that would all round to the number you were given.
A length is given as 27 m to the nearest metre. What is the degree of accuracy?
The degree of accuracy is 1 metre, since that is the size of the step the value was rounded to.
A degree of accuracy can be stated as a measure like this one, or as a place value such as 2 significant figures.
How do you find the upper and lower bounds of a rounded value?
Halve the degree of accuracy the value was rounded to, then add that to the value for the upper bound and subtract it for the lower bound.
A mass of 24 800 g given to the nearest 100 g has half of 100 as 50, so its bounds are 24 850 g and 24 750 g.
A length $l$ is given as 3.6 km correct to 1 decimal place. Complete the error interval:
$______\leq l <______$
The completed error interval is:
$3.55 \leq l < 3.65$
Both bounds are in kilometres, and both are quoted to one more decimal place than the length itself was.
True or False?
The upper bound is one of the values that the true measurement could take.
False.
An error interval is written as $L B \leq x < U B$ and so the lower bound is included but the upper bound is not.
A length of exactly 3.65 km would round to 3.7 km rather than to 3.6 km.
A number is 4500 correct to 2 significant figures. What are its bounds?
Two significant figures here means the value was rounded to the nearest 100, and half of that is 50.
The bounds are therefore 4450 and 4550, giving the error interval $4450 \leq x < 4550$ for the number.

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