60-second revision
- To compare a/b and c/d, cross-multiply: a/b > c/d when ad > bc.
- Pure recurring decimal: repeating digits over as many 9s (0.272727... = 27/99 = 3/11).
- Mixed recurring: (all digits - non-repeating digits) over 9s then 0s (0.1666... = 15/90 = 1/6).
- A fraction in lowest terms terminates only if its denominator has no prime factor other than 2 and 5.
- Multiplying decimals: total decimal places in the answer = sum of decimal places in the numbers.
- Dividing by a decimal: shift the decimal point equally in both numbers until the divisor is whole.
- 1/7 = 0.142857... and every fraction with denominator 7 uses the same six digits in rotation.
Types of fractions
| Type | Meaning | Example |
|---|---|---|
| Proper | Numerator smaller than denominator | 5/9 |
| Improper | Numerator greater than or equal to denominator | 11/4 |
| Mixed | Whole number plus a proper fraction | 2 3/4 = 11/4 |
| Equivalent | Same value, different form | 3/5 = 12/20 |
| Decimal fraction | Denominator a power of 10 | 37/100 = 0.37 |
Comparing and ordering fractions
- Same denominator: bigger numerator is bigger. Same numerator: smaller denominator is bigger (3/7 > 3/8).
- Cross-multiplication for two fractions: compare 4/9 and 5/11. 4 × 11 = 44 and 5 × 9 = 45, so 5/11 is larger.
- Decimal method for several fractions: 2/7 ~ 0.286, 3/10 = 0.3, 5/16 = 0.3125, so 2/7 < 3/10 < 5/16.
- Difference trick: when numerator and denominator differ by the same amount (2/3, 4/5, 7/8), the fraction with larger numbers is larger.
Recurring decimals to fractions
| Kind | Rule | Example |
|---|---|---|
| Pure recurring | Repeating block over the same number of 9s | 0.060606... = 6/99 = 2/33 |
| Mixed recurring | (Whole digit string - non-repeating part) over 9s for repeating digits followed by 0s for non-repeating digits | 0.1666... = (16 - 1)/90 = 15/90 = 1/6 |
| With a whole part | Keep the whole number aside | 2.444... = 2 + 4/9 = 22/9 |
Worked example: 0.2454545... has non-repeating 2 and repeating 45. Fraction = (245 - 2)/990 = 243/990 = 27/110.
Which fractions terminate?
Reduce the fraction to lowest terms first. It gives a terminating decimal only if the denominator is of the form 2^m × 5^n. Examples: 7/40 = 0.175 (40 = 2³ × 5) and 3/16 = 0.1875 terminate; 5/24 does not (24 has a factor 3). Careful: 9/30 reduces to 3/10, so it terminates even though 30 contains 3.
Operations with decimals
- Addition and subtraction: line up the decimal points (5.6 + 0.48 = 6.08).
- Multiplication: multiply as whole numbers, then count decimal places. 0.03 × 0.4 × 0.5: 3 × 4 × 5 = 60, with 2 + 1 + 1 = 4 places, giving 0.0060 = 0.006.
- Division: make the divisor whole. 2.4 ÷ 0.06 = 240 ÷ 6 = 40.
- Dividing by 0.5, 0.25 or 0.125 is the same as multiplying by 2, 4 or 8.
Operations with fractions
- Add or subtract mixed numbers by converting to improper fractions over the LCM: 1 1/2 + 2 2/3 = 9/6 + 16/6 = 25/6 = 4 1/6.
- Multiply straight across, cancelling first. Divide by multiplying with the reciprocal.
- "A fraction of the remainder" means the next fraction acts on what is left, not on the original total.
Worked example: a person keeps Rs 60,000. He spends 1/4 on a trip and 1/6 of the remainder on gifts. After the trip, 3/4 remains; gifts take away 1/6 of that, leaving 5/6 of 3/4. Money left = 60,000 × 3/4 × 5/6 = Rs 37,500.
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