How do you convert a fraction to a decimal?
Divide the numerator by the denominator. Example: 3 ÷ 4 = 0.75; 7 ÷ 8 = 0.875. Some fractions produce terminating decimals (exact answers like 0.25, 0.5, 0.75); others produce repeating decimals that never end (1/3 = 0.333…, 2/7 = 0.285714285714…). The repeating part is written with a bar notation: 1/3 = 0.3̄. In practice, repeating decimals are rounded to the needed precision.
How do you convert a decimal back to a fraction?
For terminating decimals: write the decimal digits over the appropriate power of 10, then simplify. 0.75 = 75/100 = 3/4; 0.125 = 125/1000 = 1/8. For repeating decimals: let x = 0.333…, then 10x = 3.333…, subtract: 10x − x = 3, so 9x = 3, x = 3/9 = 1/3. For mixed decimals like 0.1666…: let x = 0.1666…, 10x = 1.666…, 100x = 16.666…, 100x − 10x = 15, 90x = 15, x = 15/90 = 1/6.
Why do some fractions produce repeating decimals?
A fraction p/q (in lowest terms) produces a terminating decimal if and only if the denominator q has no prime factors other than 2 and 5 (since our decimal system is base 10 = 2 × 5). Denominators like 2, 4, 5, 8, 10, 16, 20, 25 terminate. Denominators like 3, 6, 7, 9, 11, 13 always repeat. Example: 1/7 = 0.142857142857… repeating with period 6. This is why 1/3 of a cup cannot be measured exactly on a decimal scale — it is inherently a repeating decimal.
How do you add and subtract fractions with different denominators?
Find the Least Common Multiple (LCM) of the denominators — this becomes the common denominator. Convert each fraction to an equivalent fraction with that denominator, then add/subtract numerators. Example: 1/3 + 1/4 → LCM(3,4) = 12; 1/3 = 4/12; 1/4 = 3/12; sum = 7/12. Always simplify the result by dividing by the GCF of numerator and denominator. Mixed numbers must be converted to improper fractions first.
What are common fraction-to-decimal equivalents worth memorising?
Knowing these by heart speeds up mental math significantly: 1/2 = 0.5; 1/3 ≈ 0.333; 2/3 ≈ 0.667; 1/4 = 0.25; 3/4 = 0.75; 1/5 = 0.2; 1/6 ≈ 0.167; 1/7 ≈ 0.143; 1/8 = 0.125; 3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875; 1/9 ≈ 0.111; 1/10 = 0.1; 1/12 ≈ 0.083; 1/16 = 0.0625; 1/25 = 0.04; 1/100 = 0.01 (= 1%). These are essential for cooking, engineering measurements, and mental arithmetic.