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Prealgebra / LEVEL 6 · DIFFICULTY 2/5
Decimals & Precision
Connect place value, fractions, rounding and repeating patterns.
Choose an island to read its lesson.
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.
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.
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.
Compute .
- Multiply both numbers by 100.
- Compute .
30
Round 12.096 to the nearest hundredth.
- The hundredths digit is 9; the following digit is 6.
- 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
Compute .
Compute .
Compute .
Compute .
Round 6.385 to the nearest hundredth, rounding a 5 upward.
Round 19.96 to the nearest tenth.
Four identical notebooks cost $2.35 each. A coupon removes $1.20 from the total. What is the final cost in dollars?
A 12.6-meter ribbon is cut into 0.35-meter pieces. How many equal pieces fit exactly?
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.
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.
Distinguish exact from rounded equality
A terminating decimal may equal a fraction exactly. A rounded decimal for a nonterminating fraction is only approximate.
Convert 0.056 to a reduced fraction.
- Write .
- Divide numerator and denominator by 8.
Does terminate as a base-ten decimal?
- The fraction is reduced.
- 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 terminate even though 12 has a prime factor of 3?
Preview the eight practice prompts
Write 0.45 as a reduced fraction.
Write as a decimal.
Write 0.0125 as a reduced fraction.
Does have a terminating base-ten decimal? Enter yes or no.
Does have a terminating base-ten decimal? Enter yes or no.
Is 0.333 exactly equal to ? Enter yes or no.
A scale display shows 0.875 kg exactly. Write that amount as a reduced fraction of one kilogram.
A workshop needs liter of dye. Its display uses exact decimal liters. What number should the display show?
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.
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.
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.
Convert 0.454545… to a fraction.
- Let ….
- , so .
- Reduce .
Find the 14th digit after the decimal in 0.237237237….
- The repeating block has length 3.
- 14 leaves remainder 2 when divided by 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
Write 0.777… as a reduced fraction, where 7 repeats forever.
Write 0.363636… as a reduced fraction.
What is the 20th digit after the decimal in 0.142142142…?
What is the 36th digit after the decimal in 0.583158315831…?
Write 0.1666… as a reduced fraction, where only 6 repeats.
Is 0.999… exactly equal to 1? Enter yes or no.
A machine labels positions using the repeating decimal 0.604604604…. What digit labels position 101 after the decimal?
A quantity is exactly 0.232323… liter. How many liters is eleven times that quantity? Enter a fraction.