Terminating and recurring decimal conversions · Higher
導入済み| English | 意味 |
|---|---|
| recurring decimal/rɪˈkɜːrɪŋ ˈdesɪml/ | 循環小数 |
Does the display tell the whole number?
- A calculator shows 0.333333. Is the display the exact value of one third, or only the digits that fit on the screen?
- This lesson studies recurring decimal 循环小数: A decimal with a digit or block of digits that repeats forever.
Choose the mathematical structure
- A finite decimal uses a power-of-ten denominator before simplifying. For a recurring decimal, multiply by powers of ten so the repeated tails align, then subtract. Use matching decimal places to compare numbers. A displayed rounded decimal need not be the exact fraction.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Why is 0.272727... equal to 3/11?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Multiplying by 100 shifts the two-digit repeat so the tails align; subtracting eliminates them, leaving 99x=27. Every recurring decimal is a fraction by this method.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
0.375=375/1000=3/8. For x=0.272727..., 100x=27.272727..., so 99x=27 and x=27/99=3/11. For y=0.16666..., 100y-10y=16.666...-1.666...=15, so y=15/90=1/6. In a reduced fraction, a denominator containing only factors 2 and 5 gives a terminating decimal.
Terminating and recurring decimal conversions
A finite decimal uses a power-of-ten denominator before simplifying
Compare the model with the worked case and explain one change.
Find the numerator when 0.375 is expressed over 8.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
375/1000 reduces to 3/8.
Test a tempting shortcut
- Align the recurring tails before subtracting. 0.333333 is finite and differs from 0.333333... . A non-recurring prefix needs a second power of ten; blindly dividing every digit block by 99 fails.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The finite decimal 0.333333 is exactly one third. This claim is false. Explain which definition or assumption it violates.
Find the denominator of 0.272727... in lowest terms.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
100x-x=27 gives 27/99=3/11.
The finite decimal 0.333333 is exactly one third.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Align the recurring tails before subtracting. 0.333333 is finite and differs from 0.333333... . A non-recurring prefix needs a second power of ten; blindly dividing every digit block by 99 fails.
Interpret a new situation
- AQA N10 Higher includes recurring conversion. Both tiers convert and order terminating decimals and fractions. Check a conversion by long division; 3 divided by 8 gives 0.375.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the denominator of 0.16666... in lowest terms.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
100y-10y=15 gives 15/90=1/6.
Match each decimal to its fraction in lowest terms.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Terminating decimals use a power-of-ten denominator; recurring ones need the align-and-subtract method before simplifying.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A decimal with a digit or block of digits that repeats forever. Choose the relationship, show the method, check its assumptions and interpret the result.