Fractions, Decimals and Simplification (Speed Maths)

• Vinculum (the bar over terms), Brackets (round, then curly, then square), Of (and orders: powers and roots), Division, Multiplication, Addition, Subtraction.

• Division and multiplication have equal priority, so solve them left to right. Addition and subtraction also have equal priority, so solve them left to right.

• "Of" means multiplication, and in many Indian textbooks it is solved before division and multiplication.

• Proper fraction: numerator smaller than denominator. Improper fraction: numerator larger or equal. Mixed fraction: a whole number plus a proper fraction (2 and 3/4 = 11/4).

• Addition and subtraction: use the LCM of the denominators.

• Multiplication: multiply numerators and denominators. Division: multiply by the reciprocal.

• Comparing fractions:

  • Cross-multiply (a/b against c/d: compare ad with bc).

  • If numerators are equal, the fraction with the smaller denominator is larger.

  • If denominators are equal, the larger numerator wins.

  • Convert to decimals or percentages when the fractions look unrelated.

• Recurring decimals:

  • Pure recurring: 0.454545... = 45/99 = 5/11 (put the repeating digits over as many nines).

  • Mixed recurring: 0.1666... = (16 - 1)/90 = 1/6 (all digits minus the non-repeating digits, over nines for the repeating part followed by zeros for the non-repeating part).

For other numerators, multiply: 3/8 = 3 x 12.5% = 37.5%; 5/6 = 5 x 16.67% = 83.33%.

• Square roots worth knowing: the square root of 2 is about 1.414, of 3 about 1.732, of 5 about 2.236.

• Last digit of a perfect square can only be 0, 1, 4, 5, 6 or 9, never 2, 3, 7 or 8.

• Last digit of the root from the last digit of the square: ends in 1 gives root ending 1 or 9; ends in 4 gives 2 or 8; ends in 9 gives 3 or 7; ends in 6 gives 4 or 6; ends in 5 gives 5; ends in 0 gives 0.

• Square of a number ending in 5: write n(n + 1) followed by 25. 35 squared: 3 x 4 = 12, so 1225. 85 squared: 8 x 9 = 72, so 7225.

• Squares near 50: (50 + a) squared = 2500 + 100a + a squared. 53 squared = 2500 + 300 + 9 = 2809.

• Squares near 100: (100 - a) squared = 10000 - 200a + a squared.

• Product of two numbers equidistant from a round number: (m + d)(m - d) = m squared - d squared. 47 x 53 = 50 squared - 3 squared = 2491.

• Same tens digit, last digits summing to 10: multiply the tens digit by the next number up, then append the product of the last digits. 43 x 47 = (4 x 5)(3 x 7) = 20 followed by 21 = 2021. 38 x 32 = (3 x 4)(8 x 2) = 12 followed by 16 = 1216.

• Numbers near 100: subtract each number's deficit from the other number, then multiply the deficits. 98 x 97: deficits 2 and 3; 98 - 3 = 95; 2 x 3 = 06; answer 9506.

• Multiply by 5, 25, 125: multiply by 10, 100, 1000 and divide by 2, 4, 8.

• Multiply by 9, 99, 999: multiply by 10, 100, 1000 and subtract the number.

• Multiply by 11: write the first digit, then the sum of each pair of neighbouring digits, then the last digit. 53 x 11 = 5 (5+3) 3 = 583.

• Approximation: round every number to a convenient one, calculate, and choose the nearest option. Use it whenever the options are far apart.

Example 1. Simplify 18 / 3 x 2 + 5 - 4 x 2.

• 18 / 3 = 6; 6 x 2 = 12; 4 x 2 = 8. Then 12 + 5 - 8 = 9.

Example 2. Simplify (3/4 of 240) / (5/8 of 96).

• 3/4 of 240 = 180. 5/8 of 96 = 60. 180 / 60 = 3.

Example 3. Arrange 5/7, 7/9 and 3/4 in ascending order.

• Decimals: 5/7 = 0.714, 7/9 = 0.778, 3/4 = 0.75.

• Ascending order: 5/7 \< 3/4 \< 7/9.

Example 4. Convert 0.1666... to a fraction.

• (16 - 1)/90 = 15/90 = 1/6.

Example 5. Find the square root of 7569.

• The last digit 9 means the root ends in 3 or 7. 80 squared = 6400 and 90 squared = 8100, so the root is between 80 and 90.

• 85 squared = 7225, which is less than 7569, so the root is above 85: 87. (87 squared = 7569.)

Example 6. If a + 1/a = 5, find a squared + 1/(a squared).

• Use the identity: 5 squared - 2 = 23.

• Solving addition before division because it appears first. Always follow VBODMAS and left-to-right order for equal priority.

• Forgetting that "of" means multiplication.

• Comparing fractions by looking only at the numerators or only at the denominators.

• Rounding too aggressively when two options are close.