AFCAT Numerical Ability · Decimal and Fractions

Fraction Arithmetic, Fraction Word Problems and Simplification

Work with fractions and mixed numbers, find a whole from a fraction of it, and simplify long expressions in the right order or with an algebraic shortcut.

All Decimal and Fractions notes

Why this matters

Five of the chapter's AFCAT past-year questions are here, from 2016, 2017 and 2019: two fraction word problems and three simplifications. The marks go on the order of operations ('of' first, then division and multiplication from left to right) and on dividing by the wrong fraction in a word problem.

Concept 1 of 6: Operations on fractions and mixed numbers

Mixed numbers are awkward to multiply or divide, so turn them into improper fractions first. Then multiply straight across, and divide by multiplying by the flipped fraction.

Definition

Mixed to improper: $2\tfrac13 = \tfrac{2 \times 3 + 1}{3} = \tfrac73$ .
Add or subtract over the LCM of the denominators. With mixed numbers, the whole parts can be added separately.
Multiply: tops together, bottoms together, cancelling first.
Divide: multiply by the reciprocal of the divisor.
'Of' means multiply: $\tfrac23$ of 60 is 40.

Dividing by a fraction

\dfrac ab \div \dfrac cd = \dfrac ab \times \dfrac dc

Worked example

Find

2\tfrac13 \times 1\tfrac27 \div 1\tfrac12

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Decimal and Fractions · Fractions and Simplification

4\frac{1}{5} \times 4\frac{2}{7} \div 3\frac{1}{3}

is equal to

Mixed numbers multiply only as improper fractions

2\tfrac12 \times 2\tfrac12 = \tfrac52 \times \tfrac52 = 6\tfrac14

. Multiplying the wholes and the halves apart gives

4\tfrac14

Of means times

\tfrac35

\tfrac23

\tfrac35 \times \tfrac23 = \tfrac25

, never

\tfrac35 - \tfrac23

Drill 4 more on operations on fractions and mixed numbers

Concept 2 of 6: Finding the whole from a fraction of it

If a known amount is a fraction of something, the whole is the amount divided by that fraction. When a tank goes from one fraction full to another, the amount poured in is the difference of the two fractions.

Definition

'A fraction $f$ of the whole is $A$ ' gives whole $= A \div f$ .
A level that changes from $f_1$ to $f_2$ because $A$ litres went in or out: $A$ is $(f_2 - f_1)$ of the capacity.
'One fraction of a number is $A$ more than another fraction of it': $A$ is the difference of the fractions.
Once the whole is known, any other fraction of it follows.

Whole from a part

\text{whole} = \dfrac{\text{amount}}{\text{the fraction it is (or the difference of two fractions)}}

Worked example

A drum is

\tfrac25

full. After 36 L is poured in, it is

\tfrac{7}{10}

full. Find its capacity.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2017 · AFCAT (II) 2017 — Numerical Ability · 01-310Easy

Example 2 · Decimal and Fractions · Fractions and Simplification

\frac{1}{5}

th tank is full and if 22 L fuel poured in, then

\frac{3}{4}

th tank is full. Find the capacity of tank.

Divide by the difference, not by either fraction

In the drum above,

36 \div \tfrac{7}{10}

is about 51 L and is wrong: the 36 L filled only the gap between the two levels.

Remains means the new level

'After 30 L is drawn, half remains' puts the level at

\tfrac12

. The fraction drawn out is the old level minus

\tfrac12

Drill 3 more on finding the whole from a fraction of it

Concept 3 of 6: The wrong-operation problem

Someone was told to do one thing to a number and did another. Both results are fractions of the same unknown number, so their difference is a fraction of it too, and that difference is given.

Definition

Write both results as multiples of $x$ . Dividing by $\tfrac pq$ is multiplying by $\tfrac qp$ .
Subtract the smaller multiple from the larger and set it equal to the given difference.
Solve for $x$ , then read the question again: it may want the number, or the correct result.

Wrong result against right result

\left(\dfrac{q}{p} - \dfrac{p}{q}\right) x = d \quad (\text{divided by } \tfrac pq \text{ instead of multiplying})

Worked example

A student was to multiply a number by

\tfrac45

but divided it by

\tfrac45

, and got 36 more than the right answer. Find the number.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 01-312Moderate

Example 3 · Decimal and Fractions · Fractions and Simplification

In an examination, a student was asked to find

\frac{3}{14}

of a certain number. By mistake, he found

\frac{3}{4}

of it. His answer was 150 more than the correct answer. Find the number.

Dividing by a fraction flips it

Dividing by

\tfrac45

multiplies by

\tfrac54

, so the wrong result is the larger one. Writing it as

\tfrac45 x

again makes the difference zero.

Answer what is asked

In 'multiplied by 5 instead of dividing, and got 125', the number is 25 but the correct answer is

25 \div 5 = 5

. Both are usually among the options.

Drill 2 more on the wrong-operation problem

Concept 4 of 6: Order of operations (BODMAS)

A long expression has one value only because everyone follows the same order: brackets from the inside out, then 'of', then division and multiplication, then addition and subtraction.

Definition

Brackets, innermost first: $( \; )$ , then $\{ \; \}$ , then $[ \; ]$ .
Of next: $8 \div \tfrac12$ of $4 = 8 \div 2 = 4$ , while $8 \div \tfrac12 \times 4 = 64$ .
Division and multiplication as they come, left to right.
Addition and subtraction as they come, left to right.
A minus sign in front of a bracket changes the sign of everything inside when the bracket is removed.
Decimals follow the same order: multiply and divide before you add.

Order of operations

\text{brackets} \to \text{of} \to \div \text{ and } \times \to + \text{ and } -

Worked example

Simplify

60 - [14 - \{9 - (3 + 2)\}] + 18 \div 2 \times 3

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Decimal and Fractions · Fractions and Simplification

Simplify

1035 \div \left[\left(\frac{3}{4}\right) \text{ of } (71 + 65) - 15\frac{3}{4}\right]

Division and multiplication go left to right

36 \div 6 \times 3 = 6 \times 3 = 18

. Doing the multiplication first gives

36 \div 18 = 2

A minus before a bracket flips the signs inside

20 - (8 - x) = 12 + x

, not

12 - x

Drill 12 more on order of operations (bodmas)

Concept 5 of 6: Complex and continued fractions

A fraction with fractions above and below its line is two separate sums. Finish the top, finish the bottom, then divide. A continued fraction is the same idea stacked, so start at the bottom and climb.

Definition

Simplify the numerator to one fraction and the denominator to one fraction.
Then divide: multiply the top by the reciprocal of the bottom.
Inside either part, BODMAS applies, and 'of' means multiply.
Continued fraction: work out the lowest level first, flip it, add the level above, and repeat.

A fraction over a fraction

\dfrac{a / b}{c / d} = \dfrac ab \times \dfrac dc

Worked example

Simplify

\dfrac{\tfrac12 + \tfrac34}{2\tfrac13 - \tfrac12}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 01-304Moderate

Example 5 · Decimal and Fractions · Fractions and Simplification

\cfrac{1\frac{1}{4} \div 1\frac{1}{2}}{\left(\frac{1}{15} + 1 - \frac{9}{10}\right)}

is equal to

Never cancel across a plus sign

\dfrac{6 + 8}{2} = 7

. Cancelling the 2 into the 6 alone, to get

3 + 8

, treats a sum as if it were a product.

A continued fraction is read from the bottom

1 + \cfrac{1}{1 + \cfrac13}

, the

1 + \tfrac13

at the bottom must be found first. Starting at the top adds the wrong things together.

Drill 2 more on complex and continued fractions

Concept 6 of 6: Algebraic shortcuts in simplification

When an expression has the shape of an identity, the arithmetic collapses.

a^2 - b^2

with awkward decimals becomes

(a - b)(a + b)

, usually two round numbers.

Definition

$a^2 - b^2 = (a - b)(a + b)$ .
$a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$ .
$\dfrac{a^3 - b^3}{a^2 + ab + b^2} = a - b$ and $\dfrac{a^3 + b^3}{a^2 - ab + b^2} = a + b$ .
If $a + b + c = 0$ , then $a^3 + b^3 + c^3 - 3abc = 0$ .
A common factor comes out: $ab - ac = a(b - c)$ .

Difference of squares

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

Worked example

Find

\dfrac{5.5 \times 5.5 - 2.5 \times 2.5}{0.3}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 01-302Easy

Example 6 · Decimal and Fractions · Fractions and Simplification

\frac{3.25 \times 3.20 - 3.20 \times 3.05}{0.064}

is equal to

The zero needs a + b + c = 0

For

a = 1

b = 2

c = 3

a^3 + b^3 + c^3 - 3abc = 1 + 8 + 27 - 18 = 18

, not 0. Check the sum before using the shortcut.

Look for the shape before multiplying

Squaring

5.5

and

2.5

by hand works but costs time and invites a slip. A difference of two squares, or a repeated factor in two products, is the signal to factor first.

Drill 4 more on algebraic shortcuts in simplification

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 (6)

Operations on fractions and mixed numbers
Dividing by a fraction
$\dfrac ab \div \dfrac cd = \dfrac ab \times \dfrac dc$
Finding the whole from a fraction of it
Whole from a part
$\text{whole} = \dfrac{\text{amount}}{\text{the fraction it is (or the difference of two fractions)}}$
The wrong-operation problem
Wrong result against right result
$\left(\dfrac{q}{p} - \dfrac{p}{q}\right) x = d \quad (\text{divided by } \tfrac pq \text{ instead of multiplying})$
Order of operations (BODMAS)
Order of operations
$\text{brackets} \to \text{of} \to \div \text{ and } \times \to + \text{ and } -$
Complex and continued fractions
A fraction over a fraction
$\dfrac{a / b}{c / d} = \dfrac ab \times \dfrac dc$
Algebraic shortcuts in simplification
Difference of squares
$a^2 - b^2 = (a - b)(a + b)$

Watch out for (12)

Mixed numbers multiply only as improper fractions
→ Operations on fractions and mixed numbers
Of means times
→ Operations on fractions and mixed numbers
Divide by the difference, not by either fraction
→ Finding the whole from a fraction of it
Remains means the new level
→ Finding the whole from a fraction of it
Dividing by a fraction flips it
→ The wrong-operation problem
Answer what is asked
→ The wrong-operation problem
Division and multiplication go left to right
→ Order of operations (BODMAS)
A minus before a bracket flips the signs inside
→ Order of operations (BODMAS)
Never cancel across a plus sign
→ Complex and continued fractions
A continued fraction is read from the bottom
→ Complex and continued fractions
The zero needs a + b + c = 0
→ Algebraic shortcuts in simplification
Look for the shape before multiplying
→ Algebraic shortcuts in simplification

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