Problem

Find fractions between two on the number line

On a number line from 0 to 1 divided into 12 equal parts, mark the fractions

\frac{5}{6}

\frac{11}{12}

\frac{1}{2}

, and

\frac{4}{6}

, then list every one of them that lies between

\frac{5}{12}

and

\frac{3}{4}

Fractions

Your answer

How to solve

Strategy Make a Systematic List — Rewriting every fraction over the common denominator 12 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

1STEP 1

Express the endpoints in twelfths

Write the boundaries over 12: $\frac{5}{12}$ stays $\frac{5}{12}$ , and $\frac{3}{4}$ = $\frac{9}{12}$ — so look for fractions between $\frac{5}{12}$ and $\frac{9}{12}$ .

\frac{3}{4} = \frac{9}{12}

2STEP 2

Convert the four fractions to twelfths

Rewrite each fraction with denominator 12: $\frac{5}{6}$ = $\frac{10}{12}$ , $\frac{11}{12}$ stays $\frac{11}{12}$ , $\frac{1}{2}$ = $\frac{6}{12}$ , and $\frac{4}{6}$ = $\frac{8}{12}$ .

\frac{5}{6}=\frac{10}{12}, \frac{1}{2}=\frac{6}{12}, \frac{4}{6}=\frac{8}{12}

3STEP 3

List which fall between $\frac{5}{12}$ and $\frac{9}{12}$

In twelfths, $\frac{6}{12}$ and $\frac{8}{12}$ fall inside ( $\frac{5}{12}$ , $\frac{9}{12}$ ), but $\frac{10}{12}$ and $\frac{11}{12}$ do not — so $\frac{1}{2}$ and $\frac{4}{6}$ lie between.

\frac{5}{12} \lt \frac{6}{12}, \frac{8}{12} \lt \frac{9}{12} ≤ \frac{10}{12},\frac{11}{12}

Answer

\frac{1}{2}

and

\frac{4}{6}

In twelfths the target window is

\frac{6}{12}

through

\frac{8}{12}

;

\frac{1}{2}

\frac{6}{12}

and

\frac{4}{6}

\frac{8}{12}

sit inside it, while

\frac{5}{6}

\frac{10}{12}

and

\frac{11}{12}

sit past

\frac{9}{12}

, so they are correctly excluded. The marks on the number line agree.

Takeaway

Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!

Express the endpoints in twelfths
Convert the four fractions to twelfths
List which fall between $\frac{5}{12}$ and $\frac{9}{12}$

Where next?

Level upsuggested

Compare the next digit to fill a blank in decimals Find the extreme decimal satisfying an inequality

Another one like thissuggested

10 practice problems

Step back and solidifysuggested

Compare fractions sharing numerator or denominator Compare decimals from the highest place down

2 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 10 problems

Find fractions between two on the number line

Express the endpoints in twelfths

Convert the four fractions to twelfths

List which fall between 512\frac{5}{12}125​ and 912\frac{9}{12}129​

List which fall between $\frac{5}{12}$ and $\frac{9}{12}$