2.1 Comparing and Converting Rational Numbers
Key Takeaways
- A rational number is any number expressible as a/b with integers a and b, b ≠ 0; every rational number has a decimal form that either terminates or repeats.
- A fraction in lowest terms terminates exactly when its denominator's prime factorization contains only 2s and 5s; any other prime factor forces a repeating decimal.
- Converting a repeating decimal to a fraction uses the shift-and-subtract method: multiply by 10^k where k is the period length, subtract, and solve.
- Common-denominator comparison, cross-multiplication, and benchmark reasoning against 0, 1/2, and 1 are three reliable ways to order fractions without a calculator.
- On a number line, comparing negative rationals reverses intuition: -3/4 lies to the right of -7/8 because -3/4 is the larger number.
2.1 Comparing and Converting Rational Numbers
Competency 1 skill 1 names four representations — fractions, terminating decimals, repeating decimals, percentages — plus the number line. Items ask you to move between them and to order values that arrive in mixed forms. Because you have only an on-screen scientific calculator, fluency here saves minutes you will need later.
What makes a number rational
A rational number is any number that can be written as a quotient a/b where a and b are integers and b ≠ 0. That definition immediately absorbs integers (5 = 5/1), terminating decimals (0.375 = 3/8), repeating decimals (0.333... = 1/3), and percentages (42% = 42/100).
The decisive theorem for the exam: every rational number has a decimal expansion that either terminates or eventually repeats, and every terminating or repeating decimal is rational. Non-repeating, non-terminating decimals such as π and √2 are irrational.
Which fractions terminate
Reduce the fraction to lowest terms and factor the denominator.
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| Terminating vs. Repeating - decided by the reduced denominator |
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| Denominator factors into only 2s and/or 5s -> TERMINATES |
| Any other prime factor (3, 7, 11, 13, ...) -> REPEATS |
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| 3/8 = 3/2^3 -> 0.375 terminates |
| 7/40 = 7/(2^3 * 5) -> 0.175 terminates |
| 5/12 = 5/(2^2 * 3) -> 0.41666... repeats (3 is present) |
| 2/7 = 2/7 -> 0.285714... repeats, period 6 |
| 6/15 = 2/5 reduced -> 0.4 terminates AFTER reducing |
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The last row is the classic trap. Unreduced, 6/15 shows a denominator of 15 = 3 · 5 and looks like it must repeat. Reduce first: 6/15 = 2/5 = 0.4. Always reduce before applying the test.
The reason is base-10 structure. A terminating decimal is a fraction whose denominator is a power of 10, and 10^n = 2^n · 5^n. A reduced fraction can be rewritten with a denominator of 10^n only if its denominator already divides some power of 10, which happens exactly when its only primes are 2 and 5.
Converting a repeating decimal to a fraction
Use shift-and-subtract. Let x equal the decimal, multiply by 10^k where k is the length of the repeating block, subtract, and solve.
Example: convert 0.727272... to a fraction. The repeating block "72" has length 2.
- x = 0.727272...
- 100x = 72.727272...
- 100x − x = 72.727272... − 0.727272... = 72
- 99x = 72, so x = 72/99 = 8/11
Example with a non-repeating lead: convert 0.41666... . Here "6" repeats but "41" does not.
- x = 0.41666...
- 100x = 41.666... (shift past the non-repeating part)
- 1000x = 416.666...
- 1000x − 100x = 375, so 900x = 375, and x = 375/900 = 5/12
Notice the shortcut this reveals: a period of length k over 9s gives the fraction. 0.‾3‾ = 3/9 = 1/3; 0.‾27‾ = 27/99 = 3/11; 0.‾123‾ = 123/999 = 41/333.
Percentages in both directions
Percent means "per hundred," so p% = p/100.
| Task | Method | Example |
|---|---|---|
| Percent to decimal | Divide by 100 (move point 2 left) | 6.5% = 0.065 |
| Decimal to percent | Multiply by 100 (move point 2 right) | 0.008 = 0.8% |
| Percent to fraction | Write over 100, reduce | 62.5% = 625/1000 = 5/8 |
| Fraction to percent | Divide, then multiply by 100 | 7/8 = 0.875 = 87.5% |
| Mixed percent | Convert the fraction part first | 33 1/3 % = 100/300 = 1/3 |
The most common middle-grades error, and a favorite distractor, is treating 0.8% as 0.8 or as 8%. It is 0.008. Percentages below 1% and above 100% both trip students, and items exploit both.
Ordering mixed representations
When a stem gives you a list like {5/8, 0.63, 62%, 5/9}, convert everything to one form. Decimals are usually fastest:
- 5/8 = 0.625
- 0.63 = 0.630
- 62% = 0.620
- 5/9 = 0.555...
Order: 5/9 < 62% < 5/8 < 0.63.
Three faster tools when the numbers cooperate:
- Benchmarks. Compare each value to 0, 1/2, and 1. 5/9 is just over 1/2; 7/8 is near 1; 1/50 is near 0. This alone often separates the options.
- Cross-multiplication. To compare a/b and c/d with b, d > 0, compare ad against cb. For 5/8 vs 7/11: 5 · 11 = 55 and 7 · 8 = 56, so 55 < 56 means 5/8 < 7/11.
- Same numerator reasoning. With equal numerators, the larger denominator gives the smaller value: 3/7 < 3/5 because sevenths are smaller pieces than fifths.
Negative rationals on the number line
Sign reverses the direction of "bigger denominator, smaller value" reasoning, and items lean on it hard.
Compare −3/4 and −7/8. In absolute value, 3/4 = 0.75 and 7/8 = 0.875, so 7/8 is farther from zero. On the number line −7/8 sits left of −3/4. Therefore −3/4 > −7/8.
The reliable rule: for negative numbers, the number with the smaller absolute value is the larger number. Students who compare magnitudes and stop get exactly the wrong order, which is why "−7/8 > −3/4" appears as a distractor.
Placing values on a number line also tests scale reading. If a line runs from −2 to 2 with eight equal intervals, each tick is 0.5; if it has twenty intervals, each tick is 0.2. Count intervals, not tick marks, and divide the total span by the interval count.
Which of the following fractions produces a terminating decimal?
Convert the repeating decimal 0.8333... (where only the 3 repeats) to a fraction in lowest terms.
A student is asked to order -2/3, -0.6, -5/8, and -7/12 from least to greatest and writes: -0.6, -5/8, -7/12, -2/3. What ordering is actually correct?