Rounding, Estimation & Error Intervals Flashcards

All 22 cards in this deck

What are the steps to round a number to a given place value?
e.g. round $2538$ to the nearest hundred

1. Find the digit in that place: hundreds digit is $5$
2. Look at the digit to its right: $3$, which is less than $5$, so the $5$ stays
3. Replace the following digits with zeros: $2500$

Which digit do you look at to round a number to 2 decimal places?
e.g. rounding $1.8763$ to 2 dp

The third decimal digit — round up if it is 5 or more.
Here it is $6$, so $1.8763 \to 1.88$

True or false? Rounding $4.97$ to 1 decimal place gives $4.9$

False. The next digit is $7$, so the $9$ rounds up and the answer is $5.0$

To what degree of accuracy should an answer in pounds (£) normally be given?

2 decimal places (the nearest penny).
e.g. £$14.3$ is written £$14.30$

A division gives $61.8...$ when working out the greatest number of whole items that can be bought. Do you round up or down?

Round down, to $61$ — there is not enough money for a 62nd item.

True or false? When a number is rounded to the nearest 1000, the digits after the thousands place are replaced by zeros.

True. Zeros keep the place value.
e.g. $29\,381 \to 29\,000$

What is the first significant figure of a number?

The first non-zero digit, counting from the left.
e.g. in $0.0874$ it is the $8$

What are the steps to round a number to 1 significant figure?
e.g. round $0.0874$ to 1 sf

1. Find the first non-zero digit: the $8$
2. Look at the next digit: $7$, which is 5 or more, so round the $8$ up to $9$
3. Keep the place value with zeros: $0.09$

True or false? In $0.0874$, the zeros after the decimal point count as significant figures.

False. Leading zeros are only placeholders; the first significant figure is the $8$

When rounding a large whole number to significant figures, what must you do to the digits you no longer need?

Replace them with zeros to keep the place value.
e.g. $87\,569$ to 3 sf is $87\,600$

True or false? Rounding $4.096$ to 3 significant figures gives $4.09$

False. The next digit is $6$, so the $9$ rounds up, giving $4.10$

True or false? To estimate the answer to a calculation you should work it out exactly and then round the answer.

False. You must round each value to 1 significant figure first; finding the exact answer scores no marks.

What are the steps to use approximation to check whether a given answer is reasonable?
e.g. is $\frac{595}{4.08 \times 5.32}$ equal to $27.1115$ or $271.115$?

1. Round each value to 1 sf: $600$, $4$, $5$
2. Work out the estimate: $600 \div 20 = 30$
3. Compare: $27.1115$ is closest, so that answer is correct

What are the steps to estimate a total cost and say whether it is an under- or overestimate?
e.g. 35 T-shirts costing £$4.85$ each

1. Round each value to 1 sf: $40$ and £$5$
2. Multiply: $40 \times 5 =$ £$200$
3. Both values were rounded up, so it is an overestimate — the real cost is less

In an estimated division, the top number is rounded up and the bottom number is rounded down. Is the estimate an underestimate or an overestimate?

An overestimate — you are dividing a larger number by a smaller number.

What are the steps to write an error interval for a rounded value?
e.g. $x$ rounded to 1 decimal place is $9.8$

1. Halve the rounding unit: $0.1 \div 2 = 0.05$
2. Subtract and add it: $9.75$ and $9.85$
3. Write the interval: $9.75 \le x < 9.85$

In an error interval for a rounded value, why is the upper limit written with $<$ rather than $\le$?

Because a value equal to the upper limit would round up to the next value instead.
e.g. $9.85$ rounds to $9.9$, not $9.8$

What does it mean to truncate a number to 1 decimal place?

Cut off (throw away) all digits after the first decimal place, without rounding.
e.g. $8.37$ truncated to 1 dp is $8.3$

How do you write the error interval for a truncated value?
e.g. $y$ truncated to 1 decimal place gives $8.3$

The truncated value is the lower limit and the next value at that accuracy is the upper limit: $8.3 \le y < 8.4$

True or false? If $x = 4700$ correct to 2 significant figures, the error interval is $4699.5 \le x < 4700.5$

False. To 2 sf the rounding unit is $100$, so the interval is $4650 \le x < 4750$

What are the steps to estimate a real length from a photograph using a known reference?

e.g. a man stands next to a door, and in the photo he is about three quarters as tall as the door

1. Pick an object whose real size you know: a door is about $2$ m tall.
2. Compare the unknown with it in the picture: the man is about $\frac{3}{4}$ of the door.
3. Multiply: $\frac{3}{4} \times 2 = 1.5$ m.

When estimating sizes in a picture, what real height is used for an adult person as a reference?

About $1.8$ m (any estimate from $1.5$ m to $2$ m is sensible).