AFCAT Numerical Ability · Decimal and Fractions

Place Value, and Moving Between Fractions and Decimals

Each place after the point is a tenth of the one before it, so every terminating decimal is a fraction over 10, 100 or 1000, and every such fraction is a decimal.

All Decimal and Fractions notes

Why this matters

Almost every question here is one step: name a place, read a decimal in words, or turn a fraction into a decimal and back. The marks go on counting places (six hundredths against six tenths) and on stopping before lowest terms when the options carry the unreduced fraction as well.

Concept 1 of 4: Place value after the decimal point

Moving one place to the right of the point divides by 10. So the first place counts tenths, the second hundredths, the third thousandths, and a digit is worth its own value times its place.

Definition

The value of a digit is the digit times its place. Every zero after the point still holds a place, so count each one. Between any two decimals there are infinitely many more: between $0.3$ and $0.4$ lie $0.31$ , $0.311$ , $0.3111$ and so on.

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 $\tfrac{7}{10000} = 0.0007$

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Decimal and Fractions · Decimal and Fraction Conversion

The place value of

3

0.0345

is:

A zero holds a place

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.

Drill 3 more on place value after the decimal point

Concept 2 of 4: Reading and writing decimals in words

Read the digits after the point as one whole number, then name the place of the last digit.

0.47

ends in the hundredths place, so it is forty-seven hundredths.

Definition

Decimal to words: the digits after the point, then the place of the last digit ( $0.0091$ is ninety-one ten-thousandths).
Words to decimal: the place name says how many digits go after the point (tenths 1, hundredths 2, thousandths 3, ten-thousandths 4). Fill the front with zeros.
Spoken digit by digit, $0.47$ is 'point four seven'. That is not its name in place value.

A decimal with n places

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

Worked example

Write

0.0612

in words. Then write 'three hundred five thousandths' as a decimal.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Decimal and Fractions · Decimal and Fraction Conversion

Write in decimal form: Two hundred thirty-four ten-thousandths

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.

Drill 6 more on reading and writing decimals in words

Concept 3 of 4: Writing a fraction as a decimal

If the denominator can be made 10, 100 or 1000, the numerator read off is the decimal. Otherwise divide the numerator by the denominator.

Definition

Multiply top and bottom by the number that turns the denominator into 10, 100 or 1000: $\tfrac{3}{25} = \tfrac{12}{100} = 0.12$ .
Halves, quarters and fifths are worth knowing by heart: $\tfrac12 = 0.5$ , $\tfrac14 = 0.25$ , $\tfrac34 = 0.75$ , $\tfrac15 = 0.2$ .
Eighths have three places: $\tfrac18 = 0.125$ , and each further eighth adds $0.125$ .
A fraction larger than 1 gives a whole part: $\tfrac74 = 1.75$ .

Fraction to decimal

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

Worked example

Write

\tfrac38

and

\tfrac{9}{20}

as decimals.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Decimal and Fractions · Decimal and Fraction Conversion

\frac{7}{8}

in decimal is:

The digits of the fraction are not the decimal

\tfrac34

0.75

, not

0.34

, and

\tfrac25

0.4

, not

0.25

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

Eighths need three places

\tfrac18

0.125

. Rounding it to

0.12

or reading it as

0.18

gives a wrong option.

Drill 13 more on writing a fraction as a decimal

Concept 4 of 4: Writing a decimal as a fraction in lowest terms

A decimal with two places is a number of hundredths, so write it over 100. Then cancel the common factor until none is left.

Definition

Write the digits after the point over 1 followed by as many zeros as there are places: $0.36 = \tfrac{36}{100}$ .
Divide top and bottom by their HCF to reach lowest terms ( $\tfrac{36}{100}$ , HCF 4, is $\tfrac{9}{25}$ ).
A whole part stays: $2.5 = \tfrac{25}{10} = \tfrac52$ .
A vulgar fraction is just an ordinary fraction $\tfrac ab$ , as opposed to a decimal fraction.

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}

Worked example

Write

0.36

and

0.144

as fractions in lowest terms.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Decimal and Fractions · Decimal and Fraction Conversion

Express

0.625

as a fraction:

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}

Drill 11 more on writing a decimal as a fraction in lowest terms

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)

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

Reference tables (1)

Place value after the decimal point4 rows

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 $\tfrac{7}{10000} = 0.0007$

Watch out for (7)

A zero holds a place
→ Place value after the decimal point
Places are counted from the point
→ Place value after the decimal point
Count places, not digits
→ Reading and writing decimals in words
The digits of the fraction are not the decimal
→ Writing a fraction as a decimal
Eighths need three places
→ Writing a fraction as a decimal
Lowest terms means no common factor left
→ Writing a decimal as a fraction in lowest terms
One zero in the denominator per place
→ Writing a decimal as a fraction in lowest terms

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