Round decimals and check estimates

Decimal

\(\displaystyle 6.284\)

Which digit is in the hundredths place?

Rounding chooses a nearby value

A rounded value is an approximation. The approximately-equal sign does not claim exact equality

Eight point six three lies between eight and nine. It is nearer nine, so it rounds to nine to the nearest whole number

Zero decimal places means the nearest whole number. One decimal place means the nearest tenth, two means the nearest hundredth, and three means the nearest thousandth

For these questions, an exact halfway value rounds away from zero. This convention is stated because halfway values need a tie rule

Round to the nearest whole number

Round to 0 decimal places. At an exact halfway value, round away from zero

Round 8.63 to 0 decimal places

The target is the nearest whole number

Retain the ones place

The next digit is 6, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Keep no decimal places

The slider counts hundredths from zero. Read the decimal value while moving from five point four one to five point four nine. Compare its distances from five point four and five point five

As the decimal increases, what happens to the two distances?

Inspect only the next digit to decide

Find the last place to retain, then inspect the next digit

For a next digit from zero to four, keep the retained digit. For five to nine, increase the retained magnitude by one unit, carrying when necessary

The tenths digit is two and the next digit is six, so the retained tenth increases

For hundredths, inspect the thousandths digit. Here seven makes the hundredths digit increase

Round to tenths

Round to 1 decimal place. At an exact halfway value, round away from zero

Round 7.263 to 1 decimal place

Keep exactly 1 digit after the decimal point

Retain the tenths place

The next digit is 6, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Keep one decimal place

Round to hundredths

Round to 2 decimal places. At an exact halfway value, round away from zero

Round 6.487 to 2 decimal places

Keep exactly 2 digits after the decimal point

Retain the hundredths place

The next digit is 7, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Keep two decimal places

The same method works for thousandths

Round to 3 decimal places. At an exact halfway value, round away from zero

Round 0.43864 to 3 decimal places

Keep exactly 3 digits after the decimal point

Retain the thousandths place

The next digit is 6, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Keep three decimal places

Use the stated halfway convention

A value exactly halfway between two candidates is equally close to both

This section rounds exact halfway values away from zero

For positive numbers this chooses the higher candidate. For negative numbers it chooses the more negative candidate

Round a positive midpoint

Round to 2 decimal places. At an exact halfway value, round away from zero

Round 4.125 to 2 decimal places

Keep exactly 2 digits after the decimal point

This value is exactly halfway between two candidates, so use the stated convention and round away from zero

Retain the hundredths place

The next digit is 5, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Do not drop an exact halfway digit

Round the magnitude, then restore the sign

For a negative decimal, apply the same digit rule to its positive magnitude, then restore the negative sign unless the rounded value is zero

Increasing a negative magnitude means moving farther from zero, towards a more negative value

Round a negative midpoint

Round to 1 decimal place. At an exact halfway value, round away from zero

Round -4.75 to 1 decimal place

Keep exactly 1 digit after the decimal point

Round the magnitude and restore the negative sign unless the rounded value is zero

This value is exactly halfway between two candidates, so use the stated convention and round away from zero

Retain the tenths place

The next digit is 5, which is at least 5, so increase the retained magnitude by one unit

The rounded value is an approximation, not the exact original number

Keep the sign in a hundredths answer

Carry correctly and keep the requested zeros

A retained nine may require a carry into the next place. More than one nine can carry

The final decimal zeros show the requested precision. Eight is the same numerical value, but eight point zero zero records two decimal places

A small magnitude can round to zero

Zero has no separate negative position. Keep the required number of decimal zeros

Carry across the decimal point

Round to 2 decimal places. At an exact halfway value, round away from zero

Round 7.996 to 2 decimal places

Keep exactly 2 digits after the decimal point

Retain the hundredths place

The next digit is 6, which is at least 5, so increase the retained magnitude by one unit

Keep the final decimal zeros to show the requested precision

The rounded value is an approximation, not the exact original number

Retain the final zero

Write a zero result to three decimal places

Round once from the original value

Keep the original digits until you know the final required precision. Rounding in stages can change the answer

The hundredths digit in the original value is four, so the tenths digit stays two

Rounding first to two decimal places gives six point two five. Rounding that new value to one decimal place would give six point three, which is not the direct rounding of the original value

Use the original deciding digit

Read the requested recording precision

A question or measurement record may specify how many decimal places to use. Keep the unit and follow that precision

A length record requires metres to two decimal places. A calculated length of four point seven three eight metres would be recorded as four point seven four metres

A rounded record is not the original exact value. Extra printed digits do not create extra measurement accuracy

Record a length as requested

Cord required

\(\displaystyle 7.18\,\mathrm{m}\)

A project needs seven point one eight metres of cord. The shop sells cord only in whole-metre quantities. What is the least quantity that is sufficient?

Round the inputs before estimating the computation

An estimate can check the approximate size of a calculation without evaluating the exact decimal answer

In the following tasks, round each input to the nearest whole number first, then keep the original operation

Do not calculate the exact original result and then round it. That is a different method

An estimate is approximate. A nearby result can be reasonable without being proven correct

Estimate a sum using whole numbers

Estimate by rounding each input to the nearest whole number first. Round an exact halfway input away from zero

Round each input to the nearest whole number before computing

The decimal 4.62 rounds to 5

The decimal 3.18 rounds to 3

Keep the original operation and calculate using the rounded inputs

This is an estimate, not the exact decimal calculation. A result near it may be reasonable, but agreement with an estimate does not prove correctness

Round both inputs before adding

Keep multiplication when estimating a product

Estimate by rounding each input to the nearest whole number first. Round an exact halfway input away from zero

Round each input to the nearest whole number before computing

The decimal 6.18 rounds to 6

The decimal 4.83 rounds to 5

Keep the original operation and calculate using the rounded inputs

This is an estimate, not the exact decimal calculation. A result near it may be reasonable, but agreement with an estimate does not prove correctness

Compute with the rounded factors

Reported calculation

\(\displaystyle 4.82\times5.71=2.7522\)

A calculation is reported as shown. What is the sensible conclusion after estimating with whole-number inputs?

Keep the place, purpose and approximation clear

Find the retained decimal place and inspect the next digit once

At exact halfway values use the stated convention, rounding away from zero here

Carry when needed, keep the requested final decimal zeros and restore a negative sign unless the rounded result is zero

Round once from the original value

A nearest-value estimate may be insufficient when a whole-unit purchase must cover a requirement

To estimate a computation, round the inputs first and use the original operation

A gross mismatch flags a result for checking. Agreement with an estimate does not prove exact correctness