Counting Even and Odd Numbers in a Range

Counting Even and Odd Numbers in a Range — Moosa Academy

Counting only the even numbers in a range needs one extra step beyond the ordinary count. Evens advance two at a time, so the distance between the ends has to be halved before the usual $+1$ is added.

Theorem The two rules

Count of evens $= \dfrac{\text{last even} - \text{first even}}{2} + 1$ , and count of odds $= \dfrac{\text{last odd} - \text{first odd}}{2} + 1$ .

The two formulas are identical in shape. All that changes is which endpoints you pick — the first and last even number inside the range, or the first and last odd one.

Concept Why divide by two

Take the evens

4, 6, 8, 10, 12

. Subtracting gives

12 - 4 = 8

, but that is the total distance, not the number of jumps.

Each jump covers $2$ , so there are $8 \div 2 = 4$ jumps. Adding $1$ for the starting number gives $5$ — exactly the count.

It is the same fence-post idea as before, with one difference: the posts now stand two units apart, so the distance must be divided by the step before counting.

Example Evens between 13 and 67

How many even numbers lie between $13$ and $67$ ?

"Between" excludes

13

and

67

themselves

First even inside the range:

14

Last even inside the range:

66

Apply the rule:

\dfrac{66 - 14}{2} + 1

Simplify:

\dfrac{52}{2} + 1 = 26 + 1

$27$ even numbers

Both endpoints happen to be odd here, so excluding them changes nothing for the evens. Had the range been $14$ to $66$ "between", the first even would have been $16$ .

Example Odds between 13 and 67

How many odd numbers lie between $13$ and $67$ ?

13

and

67

are both odd, and "between" excludes them

First odd inside the range:

15

Last odd inside the range:

65

Apply the rule:

\dfrac{65 - 15}{2} + 1

Simplify:

\dfrac{50}{2} + 1 = 25 + 1

$26$ odd numbers

A useful cross-check: all the numbers between $13$ and $67$ total $67 - 13 - 1 = 53$ , and $27 + 26 = 53$ . Every number is either even or odd, so the two counts must add to the whole.

Example When the ends are fractions

How many even numbers are there from $\dfrac{17}{5}$ to $\dfrac{87}{6}$ ?

Convert to decimals:

\dfrac{17}{5} = 3.4

and

\dfrac{87}{6} = 14.5

The question becomes: how many evens from

3.4

14.5

First even at or above

3.4

4

Last even at or below

14.5

14

Apply the rule:

\dfrac{14 - 4}{2} + 1 = 5 + 1

$6$ even numbers

Fractional endpoints are not a special case. Turn them into decimals, step inward to the nearest even number at each end, then use the same formula.

Summary

Both counts use $\dfrac{\text{last} - \text{first}}{2} + 1$ .
Divide by $2$ because evens and odds advance in steps of two.
Always identify the first and last qualifying number inside the range before substituting.
"Between" excludes the given endpoints; "from … to" includes them.
Check your work: evens plus odds must equal the total count of numbers in the range.
For fractional endpoints, convert to decimals and step inward to the nearest qualifying number.