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Prealgebra / LEVEL 6 · DIFFICULTY 2/5

Decimals & Precision

Connect place value, fractions, rounding and repeating patterns.

3 stages · 24 practice problems · two 6-question assessment forms

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  1. MINI QUEST 6.1Decimal Operations & RoundingRead the lesson
  2. MINI QUEST 6.2Fractions & Terminating DecimalsRead the lesson
  3. MINI QUEST 6.3Repeating Decimals & Digit PatternsRead the lesson
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STAGE 6.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Decimal Operations & Rounding

Useful preparation: Combining & Scaling Fractions

Goal: Use place value to calculate and report appropriate precision.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Align equal places

Addition and subtraction combine matching powers of ten. Trailing zeros may clarify alignment without changing the number.

2.7 = 2.70

Track the scale in products and quotients

A decimal is an integer divided by a power of ten. Multiply the integer parts, then restore the scale. In division, scale both operands equally to make the divisor whole.

0.24 × 0.3 = 24 × 3 / 1000

Round only when requested

Choose a target place and inspect the next digit. Here, a next digit of 5 or more rounds the magnitude upward. Rounding intermediate steps can change a final answer.

3.746 to hundredths is 3.75
WORKED EXAMPLE 1

Compute 4.8 ÷ 0.16.

  1. Multiply both numbers by 100.
  2. Compute 480 ÷ 16.

30

WORKED EXAMPLE 2

Round 12.096 to the nearest hundredth.

  1. The hundredths digit is 9; the following digit is 6.
  2. Rounding carries into the tenths place.

12.10

Common pitfalls

Possible mix-up: More decimal digits means a larger number.

Compare place by place; 0.9 exceeds 0.125.

Possible mix-up: Round every intermediate calculation.

Keep exact values until the requested final rounding.

Explain it to yourself

Why does multiplying both the dividend and divisor by 100 preserve a quotient?

Preview the eight practice prompts
  1. Compute 3.75 + 0.86.
  2. Compute 8.2 − 3.47.
  3. Compute 0.24 × 0.5.
  4. Compute 7.2 ÷ 0.09.
  5. Round 6.385 to the nearest hundredth, rounding a 5 upward.
  6. Round 19.96 to the nearest tenth.
  7. Four identical notebooks cost $2.35 each. A coupon removes $1.20 from the total. What is the final cost in dollars? New context
  8. A 12.6-meter ribbon is cut into 0.35-meter pieces. How many equal pieces fit exactly? New context
Open stage 6.1 in the student workspace →

STAGE 6.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Fractions & Terminating Decimals

Useful preparation: Prime Factorization · Decimal Operations & Rounding

Goal: Convert finite decimals exactly and recognize terminating fractions.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Read a decimal as counted places

A finite decimal has a denominator equal to a power of ten. Remove the decimal point for its numerator, then reduce.

0.375 = 375/1000 = 3/8

A reduced denominator predicts termination

A reduced fraction terminates exactly in base ten if its denominator has no prime factors except 2 and 5. These are the prime factors available in powers of ten.

1/40 = 1/(2³ × 5) = 0.025

Distinguish exact from rounded equality

A terminating decimal may equal a fraction exactly. A rounded decimal for a nonterminating fraction is only approximate.

1/3 ≈ 0.333, but 1/3 ≠ 0.333
WORKED EXAMPLE 1

Convert 0.056 to a reduced fraction.

  1. Write 56/1000.
  2. Divide numerator and denominator by 8.

7/125

WORKED EXAMPLE 2

Does 7/48 terminate as a base-ten decimal?

  1. The fraction is reduced.
  2. 48 has a factor of 3 as well as powers of 2.

No

Common pitfalls

Possible mix-up: Every fraction has a finite decimal.

Many rational numbers repeat forever in decimal form.

Possible mix-up: Check factors before reducing the fraction.

Cancel common factors first; they may remove a problematic denominator factor.

Explain it to yourself

Why can 3/12 terminate even though 12 has a prime factor of 3?

Preview the eight practice prompts
  1. Write 0.45 as a reduced fraction.
  2. Write 7/20 as a decimal.
  3. Write 0.0125 as a reduced fraction.
  4. Does 11/30 have a terminating base-ten decimal? Enter yes or no.
  5. Does 21/56 have a terminating base-ten decimal? Enter yes or no.
  6. Is 0.333 exactly equal to 1/3? Enter yes or no.
  7. A scale display shows 0.875 kg exactly. Write that amount as a reduced fraction of one kilogram. New context
  8. A workshop needs 3/16 liter of dye. Its display uses exact decimal liters. What number should the display show? New context
Open stage 6.2 in the student workspace →

STAGE 6.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Repeating Decimals & Digit Patterns

Useful preparation: Solving & Checking Linear Equations · Fractions & Terminating Decimals

Goal: Separate a repeating block from a finite approximation.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A remainder controls the next digit

Long division repeatedly scales a remainder by ten. A repeated remainder forces the same future digits, producing a repeating block.

1/11 = 0.090909…

Subtract aligned repeating tails

Name the decimal x and multiply by a power of ten that shifts one full repeating block. Subtract to cancel the identical infinite tails.

x = 0.2727… ⇒ 100x − x = 27

Use cycles to locate distant digits

Count positions starting with the first digit after the decimal point. Divide the requested position by the block length and use the remainder, with remainder zero indicating the last block position.

A block of length 3 repeats at positions 1, 4, 7, …
WORKED EXAMPLE 1

Convert 0.454545… to a fraction.

  1. Let x = 0.454545….
  2. 100x − x = 45, so 99x = 45.
  3. Reduce 45/99.

5/11

WORKED EXAMPLE 2

Find the 14th digit after the decimal in 0.237237237….

  1. The repeating block has length 3.
  2. 14 leaves remainder 2 when divided by 3.
  3. The second digit of 237 is 3.

3

Common pitfalls

Possible mix-up: 0.121212… equals 0.12.

The infinite tail contributes additional value.

Possible mix-up: A position divisible by the block length uses the first digit.

It uses the last digit of the block.

Explain it to yourself

Why does subtracting aligned tails work even though neither decimal ends?

Preview the eight practice prompts
  1. Write 0.777… as a reduced fraction, where 7 repeats forever.
  2. Write 0.363636… as a reduced fraction.
  3. What is the 20th digit after the decimal in 0.142142142…?
  4. What is the 36th digit after the decimal in 0.583158315831…?
  5. Write 0.1666… as a reduced fraction, where only 6 repeats.
  6. Is 0.999… exactly equal to 1? Enter yes or no.
  7. A machine labels positions using the repeating decimal 0.604604604…. What digit labels position 101 after the decimal? New context
  8. A quantity is exactly 0.232323… liter. How many liters is eleven times that quantity? Enter a fraction. New context
Open stage 6.3 in the student workspace →
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