Skip to lesson content

Lesson 8 of 11

A Peek Beyond the Point · Lesson 8 of 11

Closest Decimals

“Compare distances to a target and reason through decimal digit puzzles.”

Learning Objectives

• Find the distance between a candidate decimal and a target. • Compare candidates lying on either side of a target. • Construct close decimals under digit restrictions. • Explain why the closest candidate has the smallest distance.

Closest Decimals

Knowing which decimal is larger does not always tell us which is closer to a target. If the target is 1, the numbers 0.9 and 1.01 lie on different sides. We must compare how far each lies from 1. A number line makes this visible: closeness is about the length of the gap to the target.

Definition
Distance between two numbers

The size of the gap between their positions on the number line. Subtract the smaller number from the larger number to find this nonnegative gap.

For decimals already expressed in the same place units, we can compare gaps by counting those equal parts. The distance from 0.9 to 1 is ten hundredths. The distance from 1 to 1.01 is one hundredth. One hundredth is the smaller gap, so 1.01 is closer to 1. It does not matter that one candidate is above the target and the other is below.

Compare gaps from the target 10.911.011.11.111.01 is only one-hundredth away from 1.
Distances from 1— Compare the four candidate points with the target. The labels 1 and 1.01 are staggered because the points are very close.
Example — The nearest of four candidates

Problem
Which of 0.9, 1.1, 1.01, and 1.11 is closest to 1?

  1. 1.For 0.9, the distance is 1 − 0.9 = 0.10.
  2. 2.For 1.1, the distance is 1.1 − 1 = 0.10. For 1.01, it is 0.01. For 1.11, it is 0.11.
  3. 3.Compare the distances: 0.01 < 0.10 = 0.10 < 0.11.
  4. 4.Therefore 1.01 is closest. The candidates 0.9 and 1.1 are equally distant, but neither is the closest in this set.

The target is part of the question. Changing it can change the closest candidate even when the list remains unchanged. Before computing, write the target clearly and arrange the candidates relative to it. This also helps you decide which direction of subtraction gives a distance.

Example — Changing the target

Problem
Which of 0.9, 1.1, 1.01, and 1.11 is closest to 1.09?

  1. 1.Distances below the target: 1.09 − 0.9 = 0.19 and 1.09 − 1.01 = 0.08.
  2. 2.Distances above the target: 1.1 − 1.09 = 0.01 and 1.11 − 1.09 = 0.02.
  3. 3.The smallest distance is 0.01.
  4. 4.Therefore 1.1 is closest to 1.09. It is only one hundredth away.
Example — All candidates on one side

Problem
Which of 3.56, 3.65, and 3.099 is closest to 4?

  1. 1.All three are below 4. The largest candidate among them will have the smallest remaining gap.
  2. 2.The distances are 4 − 3.56 = 0.44, 4 − 3.65 = 0.35, and 4 − 3.099 = 0.901.
  3. 3.Compare the distances: 0.35 is the smallest.
  4. 4.Therefore 3.65 is closest to 4. Counting fractional digits would not have answered the question.

Constructing a Decimal from Given Digits

A digit puzzle limits which numbers can be formed. Work on the whole-number part first because it decides the broad interval containing the number. Then arrange the remaining digits to move towards the target. Always check whether a good candidate below the target beats the best candidate above it.

When the whole-number part is fixed below the target, making the fractional part larger brings the number closer. When the whole-number part is fixed above the target, making the fractional part smaller brings it closer. This reasoning is more efficient than listing every possible arrangement.

Example — Closest to 25

Problem
Use the digits 4, 1, 8, 2, and 5 exactly once to make a decimal with two digits before the point and three after it, as close as possible to 25.

  1. 1.The whole-number part 25 can be made. With 1, 4, and 8 left, the smallest fractional part is .148. This gives 25.148, at distance 0.148.
  2. 2.Below 25, the greatest available whole-number part is 24. The largest fraction from the remaining digits 1, 5, and 8 is .851, giving 24.851.
  3. 3.Its distance from 25 is 0.149. Compare 0.148 and 0.149.
  4. 4.Thus 25.148 is closest under this specified format. Other whole-number parts are farther away, so their fractional parts cannot improve on these two candidates.
Example — A target with unavailable digits

Problem
Use 1, 4, 0, 8, and 6 exactly once to make a decimal as close as possible to 30. At least one digit must appear on each side of the point.

  1. 1.A one-digit whole-number part is at most 8, so those candidates remain below 9 and are far from 30. A three-digit whole-number part is at least 104, also far away.
  2. 2.With two whole-number digits, the largest available part below 30 is 18. The largest fraction from 0, 4, and 6 gives 18.640, with distance 11.360.
  3. 3.The smallest available whole-number part above 30 is 40. The smallest fraction from 1, 6, and 8 gives 40.168, with distance 10.168.
  4. 4.Since 10.168 < 11.360, the closest is 40.168. The best candidate need not be below the target.

Smallest Numbers in a Required Interval

Sometimes the instruction asks for the smallest number in an interval rather than the closest to a target. The digit-order strategy still begins with the whole-number part. Leading zeros do not produce a smaller three-digit whole-number part; for instance, 014 is simply 14 and would fail a requirement to lie between 100 and 1000.

Example — Smallest between 100 and 1000

Problem
Use 1, 4, 0, 8, and 6 exactly once to make the smallest decimal between 100 and 1000.

  1. 1.The whole-number part needs three digits. Its hundreds digit must be nonzero; choose the smallest available digit, 1.
  2. 2.For the tens choose 0 and for the ones choose the smallest remaining digit, 4. The smallest possible whole-number part is 104.
  3. 3.The remaining digits 6 and 8 form the smallest fractional part in the order .68.
  4. 4.The result is 104.68. A larger whole-number part could not be rescued by a smaller fractional part.
Common mistake

The closest candidate is not automatically the smallest number, the largest number, or the one with the most matching digits. Compare actual distances, including candidates on both sides. If a puzzle specifies a digit layout, obey that layout before comparing.

Try This — Two Best Candidates

Using the digits 4, 1, 8, 2, and 5 once each, find the closest decimal to 25 that is below 25 and the closest that is above 25, using two whole-number digits and three fractional digits. Explain how you maximise or minimise the fractional parts. Compare their distances to decide the overall winner.

Check Your Understanding

Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.

Quiz

Quick check

Which of 0.8, 0.69, and 1.08 is closest to 1?

Quick check

What is the distance from 1.09 to 1.1?

Quick check

Which is closer to 5: 4.98 or 5.03?

Quick check

If the whole-number part is fixed below a target, what change to the fraction brings the number closer while it stays below?

Quick check

Which pair is equally distant from 2?

Their distances from 1 are 0.20, 0.31, and 0.08. The smallest is 0.08.

Practice Problems

Practice Problems
  1. Which is closest to 3: 2.97, 3.08, or 2.9? Show the gaps.
  2. Which is closest to 1.09: 1.08, 1.11, or 1.105?
  3. Use 4, 1, 8, 2, and 5 once each to make the closest decimal below 25, with two digits before and three after the point.
  4. Use 1, 4, 0, 8, and 6 once each to make the smallest decimal between 100 and 1000. Explain the place choices.
  5. A friend chooses 0.99 over 1.001 as the closest to 1 because 0.99 has fewer digits. Is that correct?
  6. Give two different decimals equally distant from 4, one below and one above it. Explain.

The distances are 0.03, 0.08, and 0.10. Therefore 2.97 is closest.

Key Takeaways

Key Takeaways

• Closeness means a small distance from the stated target. • Find distance by subtracting the smaller number from the larger. • Candidates on both sides of a target must be considered. • Two different candidates can be equally close. • Digit puzzles require both place-value reasoning and checking the permitted layout. • The largest fraction helps a fixed candidate below the target; the smallest helps a fixed candidate above it.

Previous · Lesson 7

Locating and Comparing Decimals

Next · Lesson 9

Addition and Subtraction of Decimals