Decimal and Fractions — AFCAT Numerical Ability Formulas and Traps

Place Value, and Moving Between Fractions and Decimals

Reading and writing decimals in words

A decimal with n places

\text{decimal} = \dfrac{\text{the digits after the point}}{10^n} \quad (n = \text{number of places})

Writing a fraction as a decimal

Fraction to decimal

\dfrac ab = \dfrac{a \times k}{b \times k} \quad \text{with } b \times k = 10, 100, 1000, \dots

Writing a decimal as a fraction in lowest terms

Decimal to fraction

0.d_1 d_2 \dots d_n = \dfrac{d_1 d_2 \dots d_n}{10^n}, \text{ then cancel the HCF}

Place value after the decimal point

Place after the point	Worth	In 6.2817
1st: tenths	$\tfrac{1}{10} = 0.1$	the 2 is worth $\tfrac{2}{10} = 0.2$
2nd: hundredths	$\tfrac{1}{100} = 0.01$	the 8 is worth $\tfrac{8}{100} = 0.08$
3rd: thousandths	$\tfrac{1}{1000} = 0.001$	the 1 is worth $\tfrac{1}{1000} = 0.001$
4th: ten-thousandths	$\tfrac{1}{10000} = 0.0001$	the 7 is worth $\tfrac{7}{10000} = 0.0007$

Common traps

A zero holds a place

In

0.07

the 7 is in the hundredths place, worth

\tfrac{7}{100}

. Skipping the zero and calling it tenths makes it ten times too big.

Places are counted from the point

The tenths digit of

52.81

is 8, the first digit after the point, not the 5 or the 2 in front of it.

Count places, not digits

Sixty-two thousandths needs three places, so it is

0.062

. Writing

0.62

gives sixty-two hundredths, ten times as much.

The digits of the fraction are not the decimal

\tfrac34

is

0.75

, not

0.34

, and

\tfrac25

is

0.4

, not

0.25

. Options are often built from the fraction's own digits.

Eighths need three places

\tfrac18

is

0.125

. Rounding it to

0.12

or reading it as

0.18

gives a wrong option.

Lowest terms means no common factor left

0.5 = \tfrac{5}{10}

is true, but it is not in lowest terms. When the question asks for simplest form and the options carry both

\tfrac{5}{10}

and

\tfrac12

, only

\tfrac12

is right.

One zero in the denominator per place

0.05

has two places, so it is

\tfrac{5}{100} = \tfrac{1}{20}

, not

\tfrac{5}{10}

.

Terminating and Recurring Decimals, and Turning Them into Fractions

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Terminating or recurring: the denominator test

The terminating test

\dfrac ab \text{ in lowest terms terminates} \;\text{means}\; b = 2^m \times 5^n

Pure recurring decimals as fractions

Pure recurring decimal

0.\overline{d_1 d_2 \dots d_n} = \dfrac{d_1 d_2 \dots d_n}{\text{as many 9s as there are repeating digits}}

Mixed recurring decimals as fractions

Mixed recurring decimal

0.a\overline{b} = \dfrac{ab - a}{90}, \quad 0.a\overline{bc} = \dfrac{abc - a}{990}, \quad 0.ab\overline{c} = \dfrac{abc - ab}{900}

Common traps

Reduce before you test

\tfrac{9}{12}

has a 3 in its denominator, but it is

\tfrac34 = 0.75

, which terminates. Cancel first, then look at the denominator.

A rounded decimal is a different number

\tfrac23 = 0.\overline{6}

. The options

0.6

,

0.66

and

0.67

are all terminating numbers close to it, and none of them equals it.

Nines, not a power of ten

0.\overline{5}

is

\tfrac59

, not

\tfrac{5}{10}

. The

\tfrac{5}{10}

option is the terminating decimal

0.5

.

The whole part is added, not written on top

2.\overline{5}

is

2 + \tfrac59

. Putting all the digits over 99, as

\tfrac{25}{99}

, gives a number less than 1.

Subtract the part that does not repeat

0.6\overline{1}

is

\tfrac{61 - 6}{90} = \tfrac{11}{18}

. Writing

\tfrac{61}{90}

skips the subtraction and is a common wrong option.

One 0 for each digit that does not repeat

0.12\overline{5}

has two digits before the repeat, so the denominator is 900, not 90 or 990.

Comparing and Ordering Decimals and Fractions

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Comparing two decimals

Padding with zeros

0.7 = 0.70 = 0.700

Ordering a list, and picking the largest or smallest

Order by padded value

0.5,\ 0.505,\ 0.05 \;\to\; 500,\ 505,\ 50 \text{ thousandths}

A decimal between two decimals

Between two decimals

a < x < b

Comparing and ordering fractions

Cross-multiplying

\dfrac ab > \dfrac cd \;\text{means}\; ad > bc

Common traps

More digits is not bigger

0.2

is greater than

0.1999

. Pad to

0.2000

and compare:

2000 > 1999

.

Trailing zeros change nothing

0.50

and

0.5

are the same number. The option 'both are equal' is right whenever only zeros at the end differ.

Read which order is asked

The options nearly always include the right list reversed. Ascending is smallest first; descending is largest first.

A longer decimal can be smaller

0.606

is less than

0.66

: padded,

606 < 660

. Ordering by length puts it in the wrong place.

Check against both ends

0.15

is above

0.1

but not below

0.12

, so it is not between

0.1

and

0.12

. An option passing one end only is the usual wrong pick.

A bigger denominator is smaller only when the numerators match

\tfrac17 < \tfrac15

, but

\tfrac67 \approx 0.857

is larger than

\tfrac45 = 0.8

.

Cross-multiply the right pairs

For

\tfrac ab

against

\tfrac cd

, compare

a \times d

with

b \times c

. Multiplying the two numerators together tells you nothing.

Adding, Subtracting, Multiplying and Dividing Decimals

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Adding and subtracting decimals

Line up the points

6.07 + 0.935 = 6.070 + 0.935

Multiplying decimals

Places in a product

\text{places in } a \times b = \text{places in } a + \text{places in } b

Dividing decimals

Clearing the divisor

\dfrac ab = \dfrac{a \times 10^n}{b \times 10^n}

Common traps

Line up the points, not the last digits

2.5 + 0.25

is

2.50 + 0.25 = 2.75

. Lining up the right-hand digits adds 25 to 25 and gives a wrong option.

A whole number still has places

10 - 3.5

is

10.0 - 3.5 = 6.5

. Treat the 10 as having a zero in the tenths.

Place the point before dropping zeros

0.5 \times 0.4

:

5 \times 4 = 20

with two places is

0.20

. Dropping the zero first and then counting two places gives

0.02

, ten times too small.

Multiplying by a decimal below 1 makes it smaller

8 \times 0.25 = 2

. If your product of a number and a decimal below 1 came out bigger than the number, the point is in the wrong place.

Move the point in both numbers

4.8 \div 0.06

is

480 \div 6 = 80

. Moving the point in the divisor only, and dividing 4.8 by 6, gives

0.8

.

Count the zeros in a small answer

0.5 \div 10 = 0.05

, not

0.005

: dividing by 10 moves the point one place, not two.

Fraction Arithmetic, Fraction Word Problems and Simplification

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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)

Common traps

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

of

\tfrac23

is

\tfrac35 \times \tfrac23 = \tfrac25

, never

\tfrac35 - \tfrac23

.

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

.

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.

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

.

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

In

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

, the

1 + \tfrac13

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

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.

Squares, Cubes and Their Roots, Including Decimals and Nested Roots

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Perfect squares and cubes, and the nearest square

Reaching the nearest square

n^2 \le N < (n + 1)^2: \quad \text{add } (n + 1)^2 - N \text{ or subtract } N - n^2

Square and cube roots of decimals

Roots of decimals

\sqrt{\dfrac{a}{10^{2n}}} = \dfrac{\sqrt a}{10^n}, \qquad \sqrt[3]{\dfrac{a}{10^{3n}}} = \dfrac{\sqrt[3]{a}}{10^n}

Nested and never-ending roots

A never-ending root

x = \sqrt{a + \sqrt{a + \dots}} \;\Rightarrow\; x^2 = a + x

Common traps

Add uses the square above, subtract the square below

For 300, the squares around it are

17^2 = 289

and

18^2 = 324

. The least number to add is 24; the least to subtract is 11. Mixing them up gives the other one.

Find the range from the number of digits

The greatest 5-digit square is

316^2 = 99856

, not

99^2

, which has only four digits.

An odd number of places has no neat root

\sqrt{0.09} = 0.3

, but

\sqrt{0.9}

is about

0.949

. The

0.3

option for

\sqrt{0.9}

is a trap.

Halve the places, do not keep them

\sqrt{0.0016} = 0.04

, with two places. Writing

0.004

keeps three and is ten times too small.

Start at the innermost root

In

\sqrt{90 + \sqrt{100}}

the

\sqrt{100}

comes first. A root does not split over a sum:

\sqrt{36 + 64} = 10

, not

6 + 8

.

Keep the positive root

x^2 - x - 2 = 0

has roots 2 and

-1

. A square root is never negative, so the answer is 2.

Decimals and Percentages: Converting, Percent Of, and Percentage Change

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Decimals to percentages and back

Decimal and percentage

x = (100x)\%, \qquad p\% = \dfrac{p}{100}

A percentage of a number

Percent of a number

p\% \text{ of } N = \dfrac{p}{100} \times N

What percentage one number is of another

Part as a percentage of the whole

\dfrac{\text{part}}{\text{whole}} \times 100\%

Percentage increase and decrease

Percentage change

\text{change}\% = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100

Common traps

Two places, not one

0.05 = 5\%

, not

50\%

or

0.5\%

. The options are usually the right answer with the point moved one place too far or too little.

Going back divides

8\% = 0.08

. Writing

0.8

moves the point only one place.

Percent of is not percent off

40\%

of 70 is 28; 70 reduced by

40\%

is 42. Read whether the question wants the part or what is left.

The whole goes underneath

'15 is what percentage of 60' is

\tfrac{15}{60} = 25\%

. Turning it round,

\tfrac{60}{15}

, gives

400\%

.

Divide by the original, not the new value

50 to 75 is a

50\%

rise (25 on 50), but 75 back to 50 is a

33\tfrac13\%

fall (25 on 75). The same gap gives a different percentage.

Up and down by the same percentage does not cancel

100 up

10\%

is 110, and 110 down

10\%

is 99, not 100.

Decimal and Fractions formulas

Place Value, and Moving Between Fractions and Decimals

Reading and writing decimals in words

Writing a fraction as a decimal

Writing a decimal as a fraction in lowest terms

Place value after the decimal point

Common traps

Terminating and Recurring Decimals, and Turning Them into Fractions

Terminating or recurring: the denominator test

Pure recurring decimals as fractions

Mixed recurring decimals as fractions

Common traps

Comparing and Ordering Decimals and Fractions

Comparing two decimals

Ordering a list, and picking the largest or smallest

A decimal between two decimals

Comparing and ordering fractions

Common traps

Adding, Subtracting, Multiplying and Dividing Decimals

Adding and subtracting decimals

Multiplying decimals

Dividing decimals

Common traps

Fraction Arithmetic, Fraction Word Problems and Simplification

Operations on fractions and mixed numbers

Finding the whole from a fraction of it

The wrong-operation problem

Order of operations (BODMAS)

Complex and continued fractions

Algebraic shortcuts in simplification

Common traps

Squares, Cubes and Their Roots, Including Decimals and Nested Roots

Perfect squares and cubes, and the nearest square

Square and cube roots of decimals

Nested and never-ending roots

Common traps

Decimals and Percentages: Converting, Percent Of, and Percentage Change

Decimals to percentages and back

A percentage of a number

What percentage one number is of another

Percentage increase and decrease

Common traps

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