Question

rounding decimals: all the possibilities
if a decimal has already been rounded, use your place value knowledge
to determine what the original number could have been.
examples
rounded to the nearest tenth → 3.6

• the smallest decimal that will round to

3.6 → 3.550

• the largest decimal that will round to

3.6 → 3.649

• so, any decimal between 3.550 - 3.649

will round to 3.6.
circle the decimals that will round to 3.6
3.548 3.63 3.67 3.573
rounded to the nearest whole → 24

• the smallest decimal that will round to

24 → 23.500

• the largest decimal that will round to

24 → 24.499

• so, any decimal between 23.500 -

24.499 will round to 24.
circle the decimals that will round to 24
24.25 23.7 24.803 23.52

1. rounded to the nearest hundredth → 7.58

• the smallest decimal that will round to 7.58 →
• the largest decimal that will round to 7.58 →
• circle all the decimals that will round to 7.58.

7.587 7.576 7.592 7.583 7.577 7.5806 7.574

1. rounded to the nearest whole → 18

• the smallest decimal that will round to 18. →
• the largest decimal that will round to 18. →
• circle all the decimals that will round to 18.

17.635 18.5 18.936 17.094 18.21 18.49 19.2

1. rounded to the nearest tenth → 4.3

• the smallest decimal that will round to 4.3. →
• the largest decimal that will round to 4.3. →
• circle all the decimals that will round to 4.3.

4.36 4.297 4.037 4.26 4.3281 4.417 4.333

Explanation:

1. Rounded to the nearest hundredth → 7.58

Step1: Find the smallest decimal

When rounding to the nearest hundredth, if the original number is \(x\), for \(x\approx7.58\) (rounded to the hundredth - place). The smallest \(x\) occurs when the thousandth - place is 5 (using the "rounding up" rule). So the smallest decimal is \(7.575\).

Step2: Find the largest decimal

The largest \(x\) occurs when the thousandth - place is 4 (using the "rounding down" rule). So the largest decimal is \(7.584\).

Step3: Circle the decimals

• For \(7.587\approx7.59\) (rounded to the hundredth - place).
• \(7.576\approx7.58\) (rounded to the hundredth - place).
• \(7.592\approx7.59\) (rounded to the hundredth - place).
• \(7.583\approx7.58\) (rounded to the hundredth - place).
• \(7.577\approx7.58\) (rounded to the hundredth - place).
• \(7.5806\approx7.58\) (rounded to the hundredth - place).
• \(7.574\approx7.57\) (rounded to the hundredth - place).

So we circle \(7.576\), \(7.583\), \(7.577\), \(7.5806\).

2. Rounded to the nearest whole → 18

Step1: Find the smallest decimal

When rounding to the nearest whole number, if the original number is \(y\), for \(y\approx18\) (rounded to the whole number). The smallest \(y\) occurs when the tenths - place is 5. So the smallest decimal is \(17.5\).

Step2: Find the largest decimal

The largest \(y\) occurs when the tenths - place is 4. So the largest decimal is \(18.499\cdots\approx18.499\) (we can write it as \(18.499\) for practical purposes).

Step3: Circle the decimals

• \(17.635\approx18\) (rounded to the whole number).
• \(18.5\approx19\) (rounded to the whole number).
• \(18.936\approx19\) (rounded to the whole number).
• \(17.094\approx17\) (rounded to the whole number).
• \(18.21\approx18\) (rounded to the whole number).
• \(18.49\approx18\) (rounded to the whole number).
• \(19.2\approx19\) (rounded to the whole number).

So we circle \(17.635\), \(18.21\), \(18.49\).

3. Rounded to the nearest tenth → 4.3

Step1: Find the smallest decimal

When rounding to the nearest tenth, if the original number is \(z\), for \(z\approx4.3\) (rounded to the tenth - place). The smallest \(z\) occurs when the hundredths - place is 5. So the smallest decimal is \(4.25\).

Step2: Find the largest decimal

The largest \(z\) occurs when the hundredths - place is 4. So the largest decimal is \(4.349\).

Step3: Circle the decimals

• \(4.36\approx4.4\) (rounded to the tenth - place).
• \(4.297\approx4.3\) (rounded to the tenth - place).
• \(4.037\approx4.0\) (rounded to the tenth - place).
• \(4.26\approx4.3\) (rounded to the tenth - place).
• \(4.3281\approx4.3\) (rounded to the tenth - place).
• \(4.417\approx4.4\) (rounded to the tenth - place).
• \(4.333\approx4.3\) (rounded to the tenth - place).

So we circle \(4.297\), \(4.26\), \(4.3281\), \(4.333\).

Answer:

1.

• The smallest decimal: \(7.575\)
• The largest decimal: \(7.584\)
• Circled decimals: \(7.576\), \(7.583\), \(7.577\), \(7.5806\)

2.

• The smallest decimal: \(17.5\)
• The largest decimal: \(18.499\)
• Circled decimals: \(17.635\), \(18.21\), \(18.49\)

3.

• The smallest decimal: \(4.25\)
• The largest decimal: \(4.349\)
• Circled decimals: \(4.297\), \(4.26\), \(4.3281\), \(4.333\)