Math 6

Finding a fraction between two fractions

Two fractions look like neighbors on the number line. Is there room for another number between them, and how do you name it?

You need

Equivalent fractionsGrades 3 to 4 Locating a fraction on the number lineGrade 3

Where you are 2 in Fractions on the number line

Fractions, ratio and percent3

Fractions15

Fractions on the number line2

Finding a fraction between two fractions

Locating a fraction on the number line

When you reach for it

On a number line marked in fifths, \displaystyle \frac{3}{5} and \displaystyle \frac{4}{5} look like next-door neighbors: one mark, then the next, nothing in between. A measuring jug looks the same way, and so does a race clock. Each one is marked only so finely. This lesson is about what lives in the gap: whether there is a number between two fractions, and how to name one.

You reach for it whenever an amount falls between two marks: water in a jug that sits above one line and below the next, a sprinter who finished after one runner and before another, a point on the number line between two labeled ones. It also leads to a surprise about the number line itself. Between any two different fractions there is always another fraction, so no two fractions are ever right next to each other.

You already know how to place a fraction on the number line: cut each whole into equal pieces, as many as the denominator says, then count that many pieces from 0 as the numerator says. Everything below uses that picture, together with Equivalent fractions and comparing fractions.

The procedure

Same denominator, with room between the numerators

When two fractions have the same denominator, they count pieces of the same size, so the one with more pieces lies further right. Between \displaystyle \frac{2}{9} and \displaystyle \frac{7}{9} lie \displaystyle \frac{3}{9}, \displaystyle \frac{4}{9}, \displaystyle \frac{5}{9} and \displaystyle \frac{6}{9}. Their numerators 3, 4, 5 and 6 are the whole numbers between 2 and 7, and each of them counts ninths.

No whole number between the numerators: cut the pieces smaller

Now take \displaystyle \frac{3}{5} and \displaystyle \frac{4}{5}. The numerators are 3 and 4, and no whole number lies between them, so with fifths alone there is nothing to name. The way out is to cut every fifth into two equal pieces. Each fifth becomes two tenths, so

\frac{3}{5} = \frac{6}{10} \qquad\qquad \frac{4}{5} = \frac{8}{10}

Multiplying the numerator and the denominator by the same number gives an equivalent fraction, which names the same point. Nothing has moved; only the marks have got finer. Now 7 lies between 6 and 8, so

\frac{3}{5} < \frac{7}{10} < \frac{4}{5}

Figure 1Cutting every fifth in two names the same points with tenths, and \displaystyle \frac{7}{10} appears between \displaystyle \frac{3}{5} and \displaystyle \frac{4}{5}.

Finer cuts make more room. Multiply both by 10 instead and you get \displaystyle \frac{30}{50} and \displaystyle \frac{40}{50}, with nine fractions between them, \displaystyle \frac{31}{50} up to \displaystyle \frac{39}{50}. When the two numerators are one apart, multiplying by 2 makes room for one fraction, multiplying by 3 for two, multiplying by 10 for nine: always one fewer than the number you multiplied by. So however many fractions you are asked for, you can find them.

Different denominators: one denominator first

Comparing numerators only works when the pieces are the same size. \displaystyle \frac{2}{3} counts thirds and \displaystyle \frac{5}{7} counts sevenths, so first rewrite both with one denominator. 21 is a multiple of both 3 and 7:

\frac{2}{3} = \frac{14}{21} \qquad\qquad \frac{5}{7} = \frac{15}{21}

That also shows which is larger: \displaystyle \frac{5}{7}. The numerators 14 and 15 are neighbors, so cut each piece in two once more: \displaystyle \frac{28}{42} and \displaystyle \frac{30}{42}, and \displaystyle \frac{29}{42} lies between them.

The halfway point

Any fraction between will do for many questions, but one choice is special: the point exactly halfway. Once the two fractions share a denominator and their numerators are an even number apart, the halfway point has the numerator exactly in the middle. \displaystyle \frac{7}{10} is one tenth above \displaystyle \frac{6}{10} and one tenth below \displaystyle \frac{8}{10}; between \displaystyle \frac{2}{9} and \displaystyle \frac{6}{9} the middle is \displaystyle \frac{4}{9}, two ninths from each end. If the numerators are an odd number apart, double the denominator first; that doubles the gap and makes it even.

For \displaystyle \frac{1}{3} and \displaystyle \frac{1}{2}, sixths give \displaystyle \frac{2}{6} and \displaystyle \frac{3}{6}, one apart. Twelfths give \displaystyle \frac{4}{12} and \displaystyle \frac{6}{12}, and the halfway point is \displaystyle \frac{5}{12}.

There is a second route. The smaller fraction is short of the halfway point by exactly as much as the larger one is over it, so the two together make two halfway points. Add them and take half:

\frac{1}{3} + \frac{1}{2} = \frac{5}{6} \qquad\qquad \text{half of } \frac{5}{6} \text{ is } \frac{5}{12}

Decimals between decimals

A decimal is a fraction with denominator 10, 100, 1000 and so on, so the same idea works. 0.3 and 0.4 have no tenths between them. Writing a zero on the end, 0.30 and 0.40, changes nothing, because 3 tenths is 30 hundredths; it is the decimal way of multiplying top and bottom by 10. Now 0.31, 0.32 and so on up to 0.39 lie between, and 0.35 is halfway.

Figure 2Zooming in on the gap between 0.3 and 0.4. In hundredths there are nine numbers inside it, and 0.35 is halfway.

When the decimals have different numbers of places, write them to the same number first, then add one more place if they are neighbors. 2.7 and 2.71 become 2.70 and 2.71, which are neighbors in hundredths, so go to thousandths: 2.700 and 2.710, with 2.705 halfway.

There is always another one

Take any two different fractions. Their halfway point is a fraction, and it lies strictly between them. Then between that new fraction and either end there is another halfway point, and between that one and the end another, with no last step. So the number line is never full of fractions: no two fractions are next to each other, and no fraction has a “next” fraction.

Figure 3Each dot is halfway from the last one to 1. The gap halves every time, but it never closes, and there is always a fraction inside it.

Whole numbers are different. Nothing whole lies between 6 and 7, and 7 really is the next whole number after 6. Fractions have no such neighbors. That is why a jug or a clock can always be marked more finely: there are always more numbers for new marks to name.

On the negative side

Negative fractions are the mirror image of the positive ones across 0. \displaystyle -\frac{3}{5} and \displaystyle -\frac{4}{5} mirror \displaystyle \frac{3}{5} and \displaystyle \frac{4}{5}, so \displaystyle -\frac{7}{10}, the mirror of \displaystyle \frac{7}{10}, lies between them. Every picture in this lesson flips over 0 the same way, so the line is never full on the left either. Negative fractions and decimals on the number line places these numbers, and Comparing and ordering rational numbers puts them in order.

Worked examples

Example 1

Same denominator, plenty of room

Name every fraction with denominator 11 that lies between \displaystyle \frac{4}{11} and \displaystyle \frac{9}{11}. Are these the only fractions between them?

Solution

Both count elevenths, so a fraction with denominator 11 lies between them exactly when its numerator lies between 4 and 9. The whole numbers there are 5, 6, 7 and 8, so the fractions are

\frac{5}{11}, \quad \frac{6}{11}, \quad \frac{7}{11}, \quad \frac{8}{11}

The ends do not count: \displaystyle \frac{4}{11} and \displaystyle \frac{9}{11} are the two edges of the gap, not points inside it.

These are not the only ones. Cut each eleventh in two: \displaystyle \frac{4}{11} = \frac{8}{22} and \displaystyle \frac{5}{11} = \frac{10}{22}, so \displaystyle \frac{9}{22} lies between \displaystyle \frac{4}{11} and \displaystyle \frac{5}{11}, and so between \displaystyle \frac{4}{11} and \displaystyle \frac{9}{11} as well. The four elevenths are only the ones a ruler marked in elevenths can show.

Example 2

Neighbors with the same denominator

(a) Find a fraction between \displaystyle \frac{5}{7} and \displaystyle \frac{6}{7}. (b) Find three.

Solution

(a) The numerators 5 and 6 have no whole number between them, so cut each seventh into two. Multiplying top and bottom by 2 keeps each fraction the same:

\frac{5}{7} = \frac{10}{14} \qquad\qquad \frac{6}{7} = \frac{12}{14}

So \displaystyle \frac{11}{14} lies between them, and it is the halfway point.

(b) Three fractions need room for three numerators. Multiplying by 4 leaves one fewer than 4, which is three: \displaystyle \frac{5}{7} = \frac{20}{28} and \displaystyle \frac{6}{7} = \frac{24}{28}, so \displaystyle \frac{21}{28}, \displaystyle \frac{22}{28} and \displaystyle \frac{23}{28} all lie between. The middle one, \displaystyle \frac{22}{28}, is \displaystyle \frac{11}{14} again, written with smaller pieces.

Example 3

Different denominators

Find a fraction between \displaystyle \frac{3}{8} and \displaystyle \frac{2}{5}.

Solution

The pieces are different sizes, so first rewrite both with one denominator. 40 is a multiple of 8 and of 5:

\frac{3}{8} = \frac{15}{40} \qquad\qquad \frac{2}{5} = \frac{16}{40}

So \displaystyle \frac{2}{5} is the larger, and the two are neighbors in fortieths. Cut each piece in two: \displaystyle \frac{30}{80} and \displaystyle \frac{32}{80}, and \displaystyle \frac{31}{80} lies between them.

Check with decimals: \displaystyle \frac{3}{8} = 0.375, \displaystyle \frac{31}{80} = 0.3875 and \displaystyle \frac{2}{5} = 0.4, in increasing order.

Example 4

Halfway on a measuring jug

Priti’s measuring jug has straight, upright sides, so equal amounts of water make equal heights. One side is marked in quarters of a liter and the other in thirds. The water is exactly halfway between the \displaystyle \frac{1}{4} liter mark and the \displaystyle \frac{1}{3} liter mark. How much water is in the jug?

Solution

Quarters and thirds share the denominator 12: \displaystyle \frac{1}{4} = \frac{3}{12} and \displaystyle \frac{1}{3} = \frac{4}{12}. The numerators are one apart, an odd gap, so double the denominator: \displaystyle \frac{6}{24} and \displaystyle \frac{8}{24}. The middle numerator is 7, so the jug holds \displaystyle \frac{7}{24} of a liter.

Check by the second route: \displaystyle \frac{1}{4} + \frac{1}{3} = \frac{7}{12}, and half of \displaystyle \frac{7}{12} is \displaystyle \frac{7}{24}.

The upright sides matter. In a jug that widens toward the top, the upper half of the gap between two marks holds more water than the lower half, so the point halfway up would not be the halfway amount.

Example 5

A photo finish

In a sprint, the photo-finish clock gives Odin a time of 11.84 seconds and Nico 11.85 seconds. Rune finished after Odin and before Nico. The clock shows thousandths of a second. What times could it show for Rune, and which is halfway?

Solution

Finishing after Odin and before Nico means Rune’s time is larger than 11.84 and smaller than 11.85. Those two are neighbors in hundredths, so add a place: 11.840 and 11.850. The thousandths between them are

11.841, \quad 11.842, \quad \dots, \quad 11.849

nine possible times, and the halfway one is 11.845.

A finer clock would show more. Between 11.840 and 11.845 lies 11.8425, and the numbers do not stop there. The clock’s last digit is where the marks stop, not where the numbers stop.

Example 6

A decimal and a fraction

Find a number between 0.7 and \displaystyle \frac{3}{4}, first as a decimal and then as a fraction.

Solution

As decimals, \displaystyle \frac{3}{4} = \frac{75}{100} = 0.75 and 0.7 = 0.70. In hundredths, 0.71, 0.72, 0.73 and 0.74 all lie between.

As fractions, \displaystyle 0.7 = \frac{7}{10} = \frac{14}{20} and \displaystyle \frac{3}{4} = \frac{15}{20}. These are neighbors in twentieths, so cut each piece in two: \displaystyle \frac{28}{40} and \displaystyle \frac{30}{40}, and \displaystyle \frac{29}{40} lies between.

The two routes agree: \displaystyle \frac{29}{40} = \frac{725}{1000} = 0.725, which sits between 0.72 and 0.73, inside the gap the decimals found.

Example 7

Is there a closest fraction to one half?

Among the fractions greater than \displaystyle \frac{1}{2}, is there one that is closest to \displaystyle \frac{1}{2}?

Solution

Try a fraction that is very close, \displaystyle \frac{501}{1000}. Its halfway point with \displaystyle \frac{1}{2} is closer still. With one denominator, \displaystyle \frac{1}{2} = \frac{500}{1000}, and the numerators are one apart, so double the denominator: \displaystyle \frac{1000}{2000} and \displaystyle \frac{1002}{2000}. The halfway point is \displaystyle \frac{1001}{2000}, which is greater than \displaystyle \frac{1}{2} but only \displaystyle \frac{1}{2000} above it, against \displaystyle \frac{1}{1000} for \displaystyle \frac{501}{1000}.

Nothing about \displaystyle \frac{501}{1000} mattered. Whatever fraction greater than \displaystyle \frac{1}{2} you choose, the halfway point between it and \displaystyle \frac{1}{2} is another fraction greater than \displaystyle \frac{1}{2} and closer to it. So there is no closest one: \displaystyle \frac{1}{2} has no next fraction.

Boundaries

The two numbers must be different. \displaystyle \frac{1}{3} and \displaystyle \frac{2}{6} are two names for one point, so nothing lies between them. Rewriting both with one denominator shows this at once.

Between means strictly between. The two ends are not between themselves. \displaystyle \frac{4}{11} does not lie between \displaystyle \frac{4}{11} and \displaystyle \frac{9}{11}.

Top and bottom are multiplied by the same number. That is what keeps the point where it was. Multiplying only the denominator, or only the numerator, makes a different number.

There are many right answers. “A fraction between” has endless correct answers; the halfway point is one of them, and it is the only one with that name.

Marks stop, numbers do not. A jug or a clock has a finest mark, and a reading from it is only as exact as that mark (Measurement error). The numbers themselves never run out.

Whole numbers and integers do not work this way. Between 6 and 7 there is no whole number, and between -3 and -2 no integer. The always-another fact belongs to fractions.

Where this goes next

Being able to squeeze a number between any two is what makes ordering rational numbers a matter of rewriting and comparing, on both sides of 0, with negative fractions and decimals filling the left half of the line.

Later, the fractions and their negatives get one name, the rational numbers, and the fact in this lesson becomes one of their defining features: they lie everywhere along the line, with no gaps between neighbors. Yet they do not cover it. Some points, such as the length of the diagonal of a square with side 1, are not fractions at all; these are the irrational numbers, and together the two kinds fill the real number line.

Exercises

Same denominator

Name every fraction with denominator 13 that lies between \displaystyle \frac{5}{13} and \displaystyle \frac{10}{13}.
Find the fraction halfway between \displaystyle \frac{4}{9} and \displaystyle \frac{5}{9}.
Find a fraction between \displaystyle \frac{2}{7} and \displaystyle \frac{3}{7}.
Find three fractions between \displaystyle \frac{5}{6} and 1.

Different denominators

Find a fraction between \displaystyle \frac{1}{4} and \displaystyle \frac{2}{7}.
Find a fraction between \displaystyle \frac{5}{6} and \displaystyle \frac{7}{8}.
Find a fraction between \displaystyle \frac{1}{5} and \displaystyle \frac{1}{4}.
Find a fraction between \displaystyle \frac{3}{10} and \displaystyle \frac{1}{3}.

The halfway point

Find the halfway point between \displaystyle \frac{1}{6} and \displaystyle \frac{1}{2}.
Find the halfway point between \displaystyle \frac{2}{5} and \displaystyle \frac{1}{2}.
Find the halfway point between 1 and \displaystyle 1\frac{1}{3}.

Decimals

Name three decimals between 0.8 and 0.9.
Find the decimal halfway between 4.06 and 4.07.
Find a decimal between 0.2 and 0.21.
Find a decimal between 0.99 and 1.

In context

Rune’s measuring jug is marked every \displaystyle \frac{1}{8} of a liter. The water is above the \displaystyle \frac{5}{8} liter mark and below the \displaystyle \frac{3}{4} liter mark. Give one amount the jug could hold.
The photo-finish clock gives Maja exactly 13.2 seconds and Priti exactly 13.3 seconds. Another runner finished after Maja and before Priti, and the clock shows hundredths. How many times could it show for that runner, and which one is halfway?
Nico’s jug is marked in fifths of a liter on one side and sixths on the other. The water is above the \displaystyle \frac{4}{5} liter mark and below the \displaystyle \frac{5}{6} liter mark. Give one amount the jug could hold.
Using the mirror picture, name a fraction between \displaystyle -\frac{2}{3} and \displaystyle -\frac{1}{3}.

Reasoning

Explain why there is no whole number between 6 and 7, but there are fractions between \displaystyle \frac{6}{1} and \displaystyle \frac{7}{1}.
Is there a fraction between \displaystyle \frac{3}{4} and \displaystyle \frac{9}{12}? Explain.
\displaystyle \frac{4}{9} and \displaystyle \frac{5}{9} become \displaystyle \frac{40}{90} and \displaystyle \frac{50}{90} when top and bottom are multiplied by 10. Explain why exactly nine fractions with denominator 90 then lie between them.
Is there a smallest fraction greater than 0? Explain.

Answers

\displaystyle \frac{6}{13}, \displaystyle \frac{7}{13}, \displaystyle \frac{8}{13}, \displaystyle \frac{9}{13}. The whole numbers between 5 and 10 are 6 to 9.
\displaystyle \frac{9}{18}, which is \displaystyle \frac{1}{2}. Doubling gives \displaystyle \frac{8}{18} and \displaystyle \frac{10}{18}.
\displaystyle \frac{5}{14}. Doubling gives \displaystyle \frac{4}{14} and \displaystyle \frac{6}{14}.
\displaystyle \frac{21}{24}, \displaystyle \frac{22}{24}, \displaystyle \frac{23}{24} (one choice). Write \displaystyle 1 = \frac{6}{6} and multiply by 4: \displaystyle \frac{20}{24} and \displaystyle \frac{24}{24}.
\displaystyle \frac{15}{56}. \displaystyle \frac{1}{4} = \frac{7}{28} and \displaystyle \frac{2}{7} = \frac{8}{28}; doubling gives \displaystyle \frac{14}{56} and \displaystyle \frac{16}{56}.
\displaystyle \frac{41}{48}. \displaystyle \frac{5}{6} = \frac{20}{24} and \displaystyle \frac{7}{8} = \frac{21}{24}; doubling gives \displaystyle \frac{40}{48} and \displaystyle \frac{42}{48}.
\displaystyle \frac{9}{40}. \displaystyle \frac{1}{5} = \frac{4}{20} and \displaystyle \frac{1}{4} = \frac{5}{20}; doubling gives \displaystyle \frac{8}{40} and \displaystyle \frac{10}{40}.
\displaystyle \frac{19}{60}. \displaystyle \frac{3}{10} = \frac{9}{30} and \displaystyle \frac{1}{3} = \frac{10}{30}; doubling gives \displaystyle \frac{18}{60} and \displaystyle \frac{20}{60}.
\displaystyle \frac{1}{3}. In sixths the ends are \displaystyle \frac{1}{6} and \displaystyle \frac{3}{6}, so the middle is \displaystyle \frac{2}{6} = \frac{1}{3}.
\displaystyle \frac{9}{20}. \displaystyle \frac{4}{10} and \displaystyle \frac{5}{10} are one apart; doubling gives \displaystyle \frac{8}{20} and \displaystyle \frac{10}{20}.
\displaystyle 1\frac{1}{6}, which is \displaystyle \frac{7}{6}. \displaystyle 1 = \frac{3}{3} and \displaystyle 1\frac{1}{3} = \frac{4}{3}; doubling gives \displaystyle \frac{6}{6} and \displaystyle \frac{8}{6}.
0.81, 0.85, 0.89 (any of 0.81 to 0.89). Write the ends as 0.80 and 0.90.
4.065. In thousandths the ends are 4.060 and 4.070.
0.205 (or any of 0.201 to 0.209). The ends are 0.200 and 0.210.
0.995 (or any of 0.991 to 0.999). The ends are 0.990 and 1.000.
\displaystyle \frac{11}{16} of a liter (one choice). \displaystyle \frac{5}{8} = \frac{10}{16} and \displaystyle \frac{3}{4} = \frac{12}{16}.
Nine times, 13.21 to 13.29 seconds; the halfway one is 13.25. The ends are 13.20 and 13.30.
\displaystyle \frac{49}{60} of a liter (one choice). \displaystyle \frac{4}{5} = \frac{24}{30} and \displaystyle \frac{5}{6} = \frac{25}{30}; doubling gives \displaystyle \frac{48}{60} and \displaystyle \frac{50}{60}.
\displaystyle -\frac{1}{2} (one choice). \displaystyle \frac{1}{2} lies between \displaystyle \frac{1}{3} and \displaystyle \frac{2}{3}, halfway in sixths, so its mirror \displaystyle -\frac{1}{2} lies between \displaystyle -\frac{2}{3} and \displaystyle -\frac{1}{3}.
Whole numbers come only in whole steps, and 7 is one step after 6, so there is no room. Fractions can cut the step into pieces: \displaystyle \frac{6}{1} = \frac{12}{2} and \displaystyle \frac{7}{1} = \frac{14}{2}, so \displaystyle \frac{13}{2} lies between, and finer cuts give more.
No. \displaystyle \frac{3}{4} = \frac{9}{12} (multiply top and bottom by 3), so they are one point, and “between” needs two different numbers.
The fractions with denominator 90 between them are those with numerators 41 to 49, the whole numbers between 40 and 50. There are nine, one fewer than 10, because multiplying by 10 turns a gap of one into a gap of ten.
No. Whatever fraction greater than 0 you choose, half of it is a smaller fraction that is still greater than 0. Halving \displaystyle \frac{1}{2} gives \displaystyle \frac{1}{4}, then \displaystyle \frac{1}{8}, with no last one.