Convert 1/2 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
CCSS 7.NS.A.2d · terminating & repeating decimals
Convert Rational Numbers to Decimals Worksheets
Convert a rational number to its decimal form and classify the result as terminating or repeating. Repeating blocks are detected from exact long-division remainders rather than a floating-point approximation.
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Rational Numbers as Decimals
Terminating & repeating decimals · Grade 7 standard · Seed 271701
1.
2.
Convert 3/2 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
3.
Convert 2/3 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
4.
Convert 8/5 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
5.
Convert 4/9 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
6.
Convert -5/4 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
7.
Convert 3/4 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
8.
Convert 4/3 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
9.
Convert 33/25 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
10.
Convert 1/3 to a decimal using division. State whether the decimal terminates or repeats.
Decimal: __________ Terminates / Repeats
Exact-construction rule: operations use normalized integer numerator/denominator pairs, number-line positions use exact ticks, repeating decimals come from remainder cycles, and money contexts use exact cents. No answer depends on binary floating-point rounding.
Terminating or repeating: reading it from the denominator
Every rational number has a decimal form that either stops or repeats. The generator finds the repeating block from the remainders of long division, not from a floating-point approximation, and marks the block with parentheses, so 0.1(6) means 0.1666 and so on.
Long division for four fractions
- 7/8: 70 ÷ 8 = 8 remainder 6, 60 ÷ 8 = 7 remainder 4, 40 ÷ 8 = 5 remainder 0. The remainder is zero, so 7/8 = 0.875 and it terminates.
- 5/12: the remainders 2, 8, 8 show that the digit 6 repeats, so 5/12 = 0.41(6).
- 2/11: 20 ÷ 11 = 1 remainder 9, then 90 ÷ 11 = 8 remainder 2, and the remainder 2 returns, so 2/11 = 0.(18).
- −3/7 has a full six-digit cycle, so −3/7 = −0.(428571).
Misconceptions about repeating decimals
- Rounding 0.666… to 0.67 and treating it as exact.
- Believing a repeating decimal is not a real fraction.
- Checking the wrong condition: a fraction in lowest terms terminates only when its denominator has no prime factors except 2 and 5.
A prediction routine
Before dividing, ask students to predict whether each fraction will terminate by factoring the denominator. Then let the long division confirm or correct the prediction. Checking each prediction against the division result is how the link between denominator and decimal gets learned.
Common Core connection
- 7.NS.A.2.d · Convert rational numbers between fractions and decimals, including repeating decimal forms.
A practical match for the skill this worksheet practises, not a claim that one worksheet teaches or assesses the whole standard. Check the official math standards.
Classroom note 1
A remainder of zero means the decimal terminates.
Classroom note 2
When a previous remainder repeats, the decimal digits from that point repeat in a cycle.
Classroom note 3
A fraction in lowest terms terminates only when its denominator's prime factors are 2s and 5s: 3/8 terminates because 8 = 2 x 2 x 2, but 5/12 repeats because 12 = 2 x 2 x 3.
Classroom note 4
The repeating block is at most (denominator minus 1) digits long; 1/7 repeats with a full six-digit cycle.
Classroom note 5
In this generator, parentheses mark the repeating block: 0.1(6) means 0.1666… .
Continue the 7.NS progression
Move to another Grade 7 rational-number skill
Questions people ask
How are repeating decimals written?
Parentheses mark the repeating block, such as 0.(3) for 0.333… or 0.1(6) for 0.1666… .
Does the tool guess from denominator patterns?
No. It performs exact remainder tracking, the same structure used in long division.
How can students predict terminating vs repeating before dividing?
Reduce the fraction, then factor the denominator. Only 2s and 5s means it terminates; any other prime factor means it repeats.
Why keep the exact repeating form instead of rounding?
Later work converts repeating decimals back to fractions, and rounding first makes that impossible. 0.1(6) converts exactly to 1/6; 0.167 does not.