CDS Mathematics · Surds, Indices and Simplification

Fractions and Decimals

Convert recurring decimals to fractions, compare fractions by cross-multiplying or by their distance from 1, and simplify decimals by spotting squares.

All Surds, Indices and Simplification notes CDS Mathematics strategy

Why this matters

Eleven PYQs, almost all EASY or MODERATE. The recurring-decimal conversion is asked in some form nearly every other year; learn the one rule for it and these are free.

Concept 1 of 3: Recurring decimals as fractions

A repeating block of

n

digits becomes a denominator of

n

nines; each non-repeating digit after the point adds a zero. The numerator is the whole digit string minus the part that does not repeat.

Definition

$0.\overline{ab} = \dfrac{ab}{99}$ , $0.\overline{a} = \dfrac a9$ .
$0.a\overline{b} = \dfrac{ab - a}{90}$ ; in general, numerator = (all digits) − (non-repeating digits), denominator = one $9$ per repeating digit followed by one $0$ per non-repeating digit.
$0.\overline{9} = 1$ .
A fraction in lowest terms terminates exactly when its denominator has no prime factor other than $2$ and $5$ .

Mixed recurring decimal

0.a\overline{b} = \dfrac{\overline{ab} - a}{90}

Worked example

Write

0.4\overline{7}

and

0.\overline{36}

as fractions.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q9Moderate

Example 1 · Surds, Indices and Simplification · Fractions and Decimals

If the points P and Q represent the real numbers

0.8\overline{3}

and

0.6\overline{2}

on the number line, then the distance between P and Q is

Read where the bar starts

0.5\overline{3}

repeats only the

3

(

= \dfrac{48}{90}

);

0.\overline{53}

repeats both digits (

= \dfrac{53}{99}

). The same four symbols give two different numbers.

Drill 4 more on recurring decimals as fractions

Concept 2 of 3: Comparing fractions and fractions of a whole

Two fractions compare by cross-multiplying. For fractions just below

1

, compare how far each is from

1

: the smaller gap is the larger fraction.

Definition

$\dfrac ab > \dfrac cd$ (positive denominators) $\iff ad > bc$ .
$\dfrac{n}{n + 1}$ grows with $n$ : $\dfrac34 < \dfrac45 < \dfrac56$ .
Adding the same $k$ to top and bottom of $\dfrac ab$ ( $a < b$ ) raises it by $\dfrac{k(b - a)}{b(b + k)}$ .
'A fraction of those who remain': take the fraction of the remainder, not of the original total.

Cross-multiplication

\dfrac ab > \dfrac cd \iff ad > bc \quad (b, d > 0)

Worked example

Which is larger,

\dfrac{5}{7}

\dfrac{8}{11}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q26Moderate

Example 2 · Surds, Indices and Simplification · Fractions and Decimals

Which one of the following fractions will have minimum change in its value if 3 is added to both the numerator and the denominator of all the fractions?

The remainder, not the total

\dfrac{11}{20}

of those who appeared' is a fraction of the students left after the absentees, not of everyone registered.

Drill 3 more on comparing fractions and fractions of a whole

Concept 3 of 3: Simplifying decimal expressions

Decimal expressions in the paper are built from squares and simple fractions in disguise. Write each decimal as a fraction or spot a square, and the arithmetic collapses.

Definition

Convert: $0.064 = \dfrac{64}{1000}$ , $6.25 = \dfrac{25}{4}$ , $4.84 = 2.2^2$ .
Spot $a^2 + 2a + 1 = (a + 1)^2$ with decimals: $0.35^2 + 0.70 + 1 = 1.35^2$ .
Under a square root, look for perfect squares in both numerator and denominator.

Square of a sum

a^2 + 2ab + b^2 = (a + b)^2

Worked example

Evaluate

\dfrac{0.25^2 + 0.5 + 1}{1.25}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q1Moderate

Example 3 · Surds, Indices and Simplification · Fractions and Decimals

What is

\sqrt{\dfrac{0.064 \times 6.25}{0.081 \times 4.84}}

equal to ?

Count decimal places when you take a root

\sqrt{0.0081} = 0.09

, not

0.9

: the root has half as many decimal places as the number.

Drill 1 more on simplifying decimal expressions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Recurring decimals as fractions
Mixed recurring decimal
$0.a\overline{b} = \dfrac{\overline{ab} - a}{90}$
Comparing fractions and fractions of a whole
Cross-multiplication
$\dfrac ab > \dfrac cd \iff ad > bc \quad (b, d > 0)$
Simplifying decimal expressions
Square of a sum
$a^2 + 2ab + b^2 = (a + b)^2$

Watch out for (3)

Read where the bar starts
→ Recurring decimals as fractions
The remainder, not the total
→ Comparing fractions and fractions of a whole
Count decimal places when you take a root
→ Simplifying decimal expressions

Test yourself on Surds, Indices and Simplification

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