AFCAT Numerical Ability · Decimal and Fractions

Comparing and Ordering Decimals and Fractions

Give every decimal the same number of places, then compare them as whole numbers. For fractions, turn them into decimals or cross-multiply.

All Decimal and Fractions notes

Why this matters

Most of this page is putting four decimals in order when they share the same digits, like 0.4, 0.44 and 0.404. The marks go on two habits: reading the longer decimal as the bigger one, and picking the reversed list when the question wanted the other order.

Concept 1 of 4: Comparing two decimals

Zeros at the end of a decimal change nothing, so

0.7 = 0.70 = 0.700

. Pad both numbers to the same length and the comparison becomes one between whole numbers.

Definition

Add zeros on the right until both have the same number of places.
Compare digit by digit from the left: tenths first, then hundredths. The first place that differs decides.
A fraction compared with a decimal: turn the fraction into a decimal first ( $\tfrac25 = 0.4$ ).

Padding with zeros

0.7 = 0.70 = 0.700

Worked example

Compare

0.709

and

0.79

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Decimal and Fractions · Comparing and Ordering

Which is greater:

0.123

0.132

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.

Drill 8 more on comparing two decimals

Concept 2 of 4: Ordering a list, and picking the largest or smallest

Pad every number in the list to the same length, read them as whole numbers, and sort those. The same step finds the largest or the smallest.

Definition

Pad all the numbers to the longest length: $0.5, 0.505, 0.05$ become $500, 505, 050$ in thousandths.
Ascending means smallest first; descending means largest first.
Check the whole option, not only its first number: two options often start the same way.

Order by padded value

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

Worked example

Arrange

0.9, 0.099, 0.909, 0.09

in ascending order.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Decimal and Fractions · Comparing and Ordering

Arrange in ascending order:

0.101, 0.11, 0.1, 0.011

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.

Drill 23 more on ordering a list, and picking the largest or smallest

Concept 3 of 4: A decimal between two decimals

Pad both ends to the same length, then any number in between is between them. If they look side by side, pad one place more: between

0.6

and

0.61

lie

0.601

0.609

Definition

Pad both ends, then check each option against both ends.
If the ends are next to each other at their length, add one more place.
There are always infinitely many decimals between two different decimals.

Between two decimals

a < x < b

Worked example

Find a decimal between

0.6

and

0.61

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Decimal and Fractions · Comparing and Ordering

Which is between

0.32

and

0.42

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.

Drill 7 more on a decimal between two decimals

Concept 4 of 4: Comparing and ordering fractions

Two fractions are hard to compare by eye when both numerator and denominator differ. Turn each into a decimal, or cross-multiply and compare the two products.

Definition

Decimals: divide and compare to two or three places ( $\tfrac37 \approx 0.429$ ).
Cross-multiplying: $\tfrac ab > \tfrac cd$ when $a \times d > b \times c$ (for positive numbers).
Same numerator: the smaller denominator is the larger fraction. Same denominator: the larger numerator is larger.

Cross-multiplying

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

Worked example

Arrange

\tfrac37, \tfrac25, \tfrac49

in ascending order.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Decimal and Fractions · Comparing and Ordering

Which of the following options has fractions in ascending order?

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.

Drill 1 more on comparing and ordering fractions

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 (4)

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

Watch out for (7)

More digits is not bigger
→ Comparing two decimals
Trailing zeros change nothing
→ Comparing two decimals
Read which order is asked
→ Ordering a list, and picking the largest or smallest
A longer decimal can be smaller
→ Ordering a list, and picking the largest or smallest
Check against both ends
→ A decimal between two decimals
A bigger denominator is smaller only when the numerators match
→ Comparing and ordering fractions
Cross-multiply the right pairs
→ Comparing and ordering fractions

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