Fractions, decimals and square roots for CDS

Fractions, decimals and roots rarely appear as long questions in CDS. They sit inside almost every other question: a percentage that is really 5/8, a mensuration answer with √3 in it, a simplification built on recurring decimals. Speed here saves minutes across the whole paper.

The aim is not to calculate more but to recognise more: to see 0.625 and think 5/8, to see 1,764 and think 42², to see √(7 + 4√3) and know it simplifies. This guide covers the methods, explains why each one works, and ends with worked questions and a practice set.

Comparing fractions

Cross-multiplication. To compare a/b and c/d (with positive denominators), compare a × d with b × c. For 5/7 and 3/4: 5 × 4 = 20 and 7 × 3 = 21, so 5/7 is smaller. This works because multiplying both fractions by the same positive number b × d does not change which is larger.

Decimals. For three or more fractions, convert each to two or three decimal places. 3/5 = 0.6, 5/8 = 0.625 and 7/12 ≈ 0.583, so 5/8 is the largest.

The gap trick. For proper fractions where the numerator is the same distance below the denominator, the larger numbers give the larger fraction: 3/4 < 5/6 < 7/8. Each is 1 minus something (1/4, 1/6, 1/8), and the smaller that something, the closer the fraction is to 1.

Recurring decimals

Pure recurring (the repetition starts right after the decimal point): write the repeating block over as many 9s as it has digits.

• 0.333… = 3/9 = 1/3
• 0.4545… = 45/99 = 5/11

Why it works: let x = 0.4545…. Then 100x = 45.4545…. Subtract: 99x = 45, so x = 45/99.

Mixed recurring (some digits before the repetition starts): take all the digits up to the end of the first repeating block, subtract the non-repeating part, and divide by as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits.

• 0.1666… (6 repeating) = (16 − 1)/90 = 15/90 = 1/6
• 0.2353535… (35 repeating) = (235 − 2)/990 = 233/990

Square roots

By prime factorisation. Pair the primes and take one from each pair. 1,764 = 2² × 3² × 7², so √1,764 = 2 × 3 × 7 = 42.

By the unit digit. A perfect square's last digit narrows the root's last digit:

A square never ends in 2, 3, 7 or 8, which eliminates options instantly. For √15,129: the root ends in 3 or 7, and 120² = 14,400 while 125² = 15,625, so the root lies between 120 and 125. It must be 123.

Roots of decimals. Count decimal places; the root has half as many. √0.0049 = 0.07, and √0.000169 = 0.013. If the number of decimal places is odd, add a zero first: √0.9 = √0.90, which is not a perfect square, so do not expect a neat answer.

Cube roots of perfect cubes

Cubes of 0 to 9 end in 0, 1, 8, 7, 4, 5, 6, 3, 2, 9. Every last digit appears exactly once, so a perfect cube's last digit fixes its root's last digit. Only 2 and 8, and 3 and 7, swap; the others stay the same.

To find ∛2,197: the last digit 7 gives a root ending in 3. Strike off the last three digits, leaving 2. The largest cube not above 2 is 1³, so the first digit is 1. The root is 13.

Surds

• Simplify by pulling out square factors: √50 = 5√2, √18 = 3√2, so √50 + √18 = 8√2. Only like surds add.
• Rationalise by multiplying by the conjugate, using (a + b)(a − b) = a² − b²: 1/(√3 + √2) = (√3 − √2)/(3 − 2) = √3 − √2.
• Compare sums of surds by squaring. √7 + √3 and √6 + √4 both square to 10 plus something: 2√21 and 2√24. So √6 + √4 is larger.
• Nested surds: √(a + 2√b) = √x + √y when x + y = a and xy = b.

Fraction–decimal pairs worth knowing

These let you switch between fractions and percentages instantly: 37.5% is 3/8 and 62.5% is 5/8. Also learn squares up to 30 and cubes up to 15. They pay back in almost every paper.

Worked questions

Question 1: Arrange 5/7, 7/9 and 3/4 in ascending order.

• 5/7 against 3/4: 5 × 4 = 20 and 7 × 3 = 21, so 5/7 < 3/4.
• 3/4 against 7/9: 3 × 9 = 27 and 4 × 7 = 28, so 3/4 < 7/9.
• Order: 5/7 < 3/4 < 7/9.

Question 2: Express 0.2353535… (35 repeating) as a fraction.

• Digits up to the end of the first repeating block: 235. Non-repeating part: 2.
• Two repeating digits and one non-repeating digit give the denominator 990.
• (235 − 2)/990 = 233/990.

Question 3: Find 0.444… + 0.3636… as a fraction.

• 0.444… = 4/9 and 0.3636… = 36/99 = 4/11.
• 4/9 + 4/11 = (44 + 36)/99 = 80/99.

Question 4: Find the cube root of 1,03,823.

• Last digit 3, so the root ends in 7.
• Strike off the last three digits, leaving 103. Since 4³ = 64 and 5³ = 125, the first digit is 4.
• Root = 47. Check: 47² = 2,209 and 2,209 × 47 = 1,03,823.

Question 5: Simplify √(7 + 4√3).

• Write 4√3 as 2√12, so you need x + y = 7 and xy = 12.
• x = 4, y = 3, so the answer is √4 + √3 = 2 + √3.
• Check: (2 + √3)² = 4 + 3 + 4√3 = 7 + 4√3.

Question 6: Given √2 = 1.414, find the value of 1/(√2 − 1).

• Rationalise first: 1/(√2 − 1) × (√2 + 1)/(√2 + 1) = (√2 + 1)/(2 − 1) = √2 + 1.
• Value = 2.414. Dividing 1 by 0.414 directly would take far longer.

Practice set

1. Express 0.777… as a fraction.
2. Find √0.0081.
3. Simplify √72.
4. Rationalise 1/(√5 − 2).
5. Express 0.1222… (2 repeating) as a fraction.
6. Find the cube root of 39,304.
7. Simplify √(5 − 2√6).
8. Which is larger: √11 − √10 or √10 − 3?

Answers

1. 7/9. One repeating digit over one 9.
2. 0.09. √81 = 9, and four decimal places become two.
3. 6√2. 72 = 36 × 2.
4. √5 + 2. Multiply by (√5 + 2); the denominator is 5 − 4 = 1.
5. 11/90. (12 − 1)/90.
6. 34. Last digit 4 gives a root ending in 4. Striking off 304 leaves 39, and 3³ = 27 ≤ 39 < 64, so the first digit is 3. Check: 34³ = 39,304.
7. √3 − √2. You need x + y = 5 and xy = 6, so x = 3 and y = 2. Check: (√3 − √2)² = 5 − 2√6.
8. √10 − 3. Rationalise each: √11 − √10 = 1/(√11 + √10) and √10 − 3 = 1/(√10 + √9). The second has the smaller denominator, so it is the larger number.

What to do next

• Learn the fraction–decimal table and squares to 30 until you can recall them without thinking
• Practise ten recurring-decimal conversions and ten square-root checks by unit digit
• Solve the surd and root questions from the last five CDS papers
• Use these skills straight away in simplification, where they appear most