math.wiki
Number › Rational numbers › Comparing and ordering rational numbers
Open on the map
Math 6 · 6.NS.C.7 · 6.NS.C.7B

Comparing and ordering rational numbers

A golf average of one and a half under par, a lake one and a quarter meters below its usual mark, a fraction beside a decimal beside a percent. When numbers arrive in different forms and on both sides of zero, how do you tell which is greater?

You need
Converting among fractions, decimals and percentsMath 6Comparing fractionsGrades 3 to 4Placing and comparing integers on a number lineMath 6
Where you are 10 in Rational numbers
Number9
Rational numbers10
▴ 2 aboveConverting between fractions and decimalsWhich fractions give terminating decimals
Comparing and ordering rational numbers
Operations with rational numbersTerminating and repeating decimals▾ Show all 10 (3 more)

When you reach for it

Numbers that describe one situation often arrive in different forms. A golf average might be \displaystyle -1\frac{1}{2} for one player and -1.3 for another. A lake gauge might read \displaystyle \frac{9}{10} of a meter above the usual mark one month and 1.15 meters below it another. A survey might report \displaystyle \frac{3}{8} of one group, 0.42 of another and 38\% of a third. To say which is greater, or to put a whole list in order, the numbers have to be brought onto one line or into one form.

Each piece of that is already in hand. Integers are compared by their places on the line (Placing and comparing integers on a number line), negative fractions and decimals have their own places on it (Negative fractions and decimals on the number line), fractions are compared with each other (Comparing fractions) and decimals with each other (Comparing and ordering decimals), and a fraction, a decimal and a percent can be turned into one another (Converting among fractions, decimals and percents). This lesson joins them, so that any two rational numbers can be compared whatever their form and whatever their sign.

It runs over two days. Part 1 compares pairs and orders lists. Part 2 writes and reads inequality statements about real situations, where “greater” has to be turned back into words such as “higher” or “better”.

The procedure

Part 1: comparing and ordering

The one rule. On a number line that runs left to right, of two different numbers the one further right is the greater. The statement -2.3 < 0.55 says that -2.3 lies to the left of 0.55. This is the rule already used for integers, unchanged. It holds for every rational number because every rational number has exactly one point on the line. Everything below is a way of finding out which point is further right without drawing the line to perfect scale.

Changing the form does not move the point. \displaystyle \frac{3}{4}, 0.75 and 75\% are three names for one point: 75\% means \displaystyle \frac{75}{100}, which is 0.75, which is \displaystyle \frac{3}{4}. So it is always allowed to rewrite numbers into whichever form is easiest to compare, compare them, and then give the answer in the forms you were given.

Sort by sign first. Every negative number lies left of 0 and every positive number lies right of it, so every negative number is less than every positive number, however each is written: \displaystyle -35 < \frac{1}{100}. When the signs differ, nothing more is needed.

Then look at the integers on either side. Each number that is not itself an integer lies between two neighboring integers, found as in Negative fractions and decimals on the number line. If two numbers lie in different gaps, the gaps decide: everything between -1 and 0 is right of everything between -2 and -1. A finer comparison is needed only when two numbers share a gap.

Same sign, same gap: put them in one form.

  • Two decimals: give them the same number of decimal places (0.6 = 0.60) and compare place by place from the left, as in Comparing and ordering decimals.
  • Two fractions: rename them over a common denominator (Comparing by finding a common denominator). Pieces of the same size compare by their count.
  • A fraction and a decimal: write the fraction as a decimal, or the decimal as a fraction. When the denominator goes evenly into 10, 100, 1000 or a larger power of ten, the decimal is exact: \displaystyle \frac{5}{8} = \frac{625}{1000} = 0.625. A fraction such as \displaystyle \frac{2}{3} has a decimal that never ends (Terminating and repeating decimals), and there the fraction route is the exact one: \displaystyle 0.7 = \frac{7}{10}.
  • A percent: write it as hundredths, \displaystyle 38\% = \frac{38}{100} = 0.38.

Two negatives: compare their opposites, then turn the answer round. Take 0.3 and 0.8. The second is further right, so 0.3 < 0.8. Now fold the line at 0. The fold carries 0.3 to -0.3 and 0.8 to -0.8, and a fold swaps left and right, so the partner of the number that was further right is now further left.

−1 0 1 −0.8 −0.3 0.3 0.8
Figure 1Folding at 0 carries 0.3 and 0.8 to -0.3 and -0.8. On the right 0.8 is further right; on the left its partner -0.8 is further left, so -0.8 < -0.3.
0.3 < 0.8 \quad\text{so}\quad -0.8 < -0.3

In words: of two negative numbers, the one further from zero is the smaller. So “greater” and “further from zero” are different questions. On the positive side they happen to agree, and on the negative side they disagree; distance from zero is the subject of Absolute value.

Ordering a list is comparing many times. Split the list into negatives, zero and positives. The negatives come first, then zero, then the positives, so only the numbers inside each group need comparing. “Least to greatest” reads the points from left to right, and “greatest to least” from right to left. Two names for the same number sit at the same point, and are joined with =.

Part 2: inequality statements in situations

One fact, two ways round. The statements

-\frac{3}{4} > -1.2 \qquad\text{and}\qquad -1.2 < -\frac{3}{4}

say the same thing: \displaystyle -\frac{3}{4} lies further right. The symbol opens toward the greater number and points at the lesser. Read aloud, the first is “negative three quarters is greater than negative one point two”.

What “further right” means depends on the situation. A signed number in a situation is measured from a zero that the situation chooses, with one direction counted as positive. Before reading an inequality, settle two questions: what does 0 stand for, and which way is positive? Then “greater” can be put back into the situation’s own words.

  • A lake. Zero is the usual water mark and positive is above it. \displaystyle -\frac{3}{4} > -1.2 says that water three quarters of a meter below the mark stands higher than water 1.2 meters below it. Greater means higher.
  • Temperatures. Zero is 0 degrees and positive is warmer. \displaystyle -1.2 < -\frac{3}{4} says that -1.2 degrees is colder than \displaystyle -\frac{3}{4} of a degree. Less means colder.
  • Golf. A score against par counts strokes taken compared with par, the number of strokes the course expects. Zero is par, and positive means strokes over par. In golf the fewest strokes win. -4 < -1 says a round four under par took fewer strokes than a round one under, so it was the better round. Less means better.

The number line has no idea which end is good. The same symbol says “lower” for the lake, “colder” for a temperature and “better” for golf, and only the situation supplies the word.

Writing a statement from words goes the other way. Turn each quantity into a signed number from the situation’s zero, then decide which is further right. “The lake was lower in March than in May” becomes March’s level < May’s level, with the two numbers put in their places.

Worked examples

Examples 1 to 4 belong to Part 1 and Examples 5 and 6 to Part 2.

Example 1

Pairs in different forms

Write < or > between each pair: (a) 0.6 and \displaystyle \frac{5}{8} \qquad (b) \displaystyle -1\frac{1}{5} and -0.9 \qquad (c) \displaystyle -\frac{2}{3} and -0.7

Solution

(a) Both are positive and both lie between 0 and 1, so they need one form. Eight goes evenly into 1000, so \displaystyle \frac{5}{8} = \frac{625}{1000} = 0.625. Write 0.6 with three places too, as 0.600. The tenths agree; in the hundredths, 0 < 2. So 0.625 is further right, and in the forms given, \displaystyle 0.6 < \frac{5}{8}.

(b) Both are negative, but they sit in different gaps. \displaystyle -1\frac{1}{5} is a fifth past -1, so it lies between -2 and -1, while -0.9 lies between -1 and 0. Every number in the gap from -1 to 0 is right of every number in the gap from -2 to -1, so \displaystyle -1\frac{1}{5} < -0.9.

(c) Both are negative and both lie between -1 and 0, so compare their opposites, \displaystyle \frac{2}{3} and 0.7. The decimal for \displaystyle \frac{2}{3} never ends, so write 0.7 as a fraction instead: \displaystyle 0.7 = \frac{7}{10}. Over the common denominator 30, \displaystyle \frac{2}{3} = \frac{20}{30} and \displaystyle \frac{7}{10} = \frac{21}{30}. So \displaystyle \frac{2}{3} < 0.7, and 0.7 is further from zero. On the negative side that makes -0.7 further left:

-0.7 < -\frac{2}{3}

which is the same as \displaystyle -\frac{2}{3} > -0.7.

Example 2

A fraction, a decimal and two percents

Put \displaystyle \frac{2}{5}, 0.42, 38\% and \displaystyle \frac{3}{8} in order from least to greatest.

Solution

All four are positive and all lie between 0 and 1, so put them in one form. Decimals work well here, since every denominator goes evenly into 1000. \displaystyle \frac{3}{8} needs three places, so give all four three places:

\begin{aligned} \frac{2}{5} &= \frac{4}{10} = 0.400 \\ 0.42 &= 0.420 \\ 38\% &= \frac{38}{100} = 0.380 \\ \frac{3}{8} &= \frac{375}{1000} = 0.375 \end{aligned}

The tenths split them into two pairs: 0.400 and 0.420 have 4 tenths, 0.380 and 0.375 have 3. In the lower pair the hundredths decide, 7 < 8, so 0.375 < 0.380. In the upper pair, 0 < 2, so 0.400 < 0.420. Back in the forms given:

\frac{3}{8} < 38\% < \frac{2}{5} < 0.42

\displaystyle \frac{3}{8} and 38\% are written with the same two digits, but they are different numbers: \displaystyle \frac{3}{8} is 37.5 hundredths and 38\% is 38 hundredths. Only the rewriting shows which is greater.

Example 3

Ordering a list with both signs

Put \displaystyle \frac{3}{4}, -2.3, 0, \displaystyle -\frac{5}{8}, \displaystyle -2\frac{1}{2} and 0.55 in order from least to greatest.

Solution

Sort by sign first. The negatives are -2.3, \displaystyle -\frac{5}{8} and \displaystyle -2\frac{1}{2}; then comes 0; the positives are \displaystyle \frac{3}{4} and 0.55.

Among the negatives, \displaystyle -\frac{5}{8} lies between -1 and 0, while -2.3 and \displaystyle -2\frac{1}{2} both lie between -3 and -2. So \displaystyle -\frac{5}{8} is the greatest of the three. For the other two, compare their opposites: \displaystyle 2\frac{1}{2} = 2.5 and 2.5 > 2.3, so \displaystyle -2\frac{1}{2} is further from zero and further left, and \displaystyle -2\frac{1}{2} < -2.3.

Among the positives, \displaystyle \frac{3}{4} = 0.75, and 0.55 < 0.75.

Joining the groups, negatives first:

-2\frac{1}{2} < -2.3 < -\frac{5}{8} < 0 < 0.55 < \frac{3}{4}
−3 −2 −1 0 1
-2\tfrac{1}{2}
 −2.3
-\tfrac{5}{8}
 0.55
\tfrac{3}{4}
Figure 2The six numbers of Example 3 on one line. Read left to right, they are in order from least to greatest.
Example 4

When rounding hides the answer

Which is greater, \displaystyle -\frac{5}{9} or -0.56?

Solution

Both are negative and both lie between -1 and 0, so compare the opposites, \displaystyle \frac{5}{9} and 0.56. The decimal for \displaystyle \frac{5}{9} never ends: it is 0.555\ldots with the fives going on forever. Rounded to two places it would read 0.56, and the two numbers would look equal. They are not, so compare exactly instead. \displaystyle 0.56 = \frac{56}{100}, and over the common denominator 900,

\frac{5}{9} = \frac{500}{900} \qquad\text{and}\qquad \frac{56}{100} = \frac{504}{900}

So \displaystyle \frac{5}{9} < 0.56. Now turn the answer round: 0.56 is further from zero, so -0.56 is further left, and \displaystyle -0.56 < -\frac{5}{9}. The greater number is \displaystyle -\frac{5}{9}.

Example 5

Golf averages against par

Over their last ten rounds, five golfers averaged these scores against par, where a negative score means under par: Sami -1.3, Tess \displaystyle -1\frac{1}{2}, Ugne \displaystyle \frac{3}{10}, Yael \displaystyle -\frac{4}{5} and Zayn -1.1.

(a) List the golfers from best average to worst. (b) Write an inequality comparing Sami’s average with Tess’s, and say what it means. (c) Write “Zayn’s average was better than Yael’s” as an inequality.

Solution

(a) Zero is par and positive means strokes over par, so the best average is the least number. Put them all in decimals: Sami -1.3, Tess -1.5, Ugne 0.3, Yael -0.8, Zayn -1.1. Ugne is the only one above par, so Ugne comes last. For the four negatives compare their opposites, 1.3, 1.5, 0.8 and 1.1: the largest opposite belongs to the least number.

-1\frac{1}{2} < -1.3 < -1.1 < -\frac{4}{5} < \frac{3}{10}

From best to worst: Tess, Sami, Zayn, Yael, Ugne.

−2 −1 0 1 Tess Sami Zayn Yael Ugne
Figure 3The five golf averages of Example 5. Further left means fewer strokes, which in golf is better.

(b) \displaystyle -1\frac{1}{2} < -1.3, or the same fact the other way round, \displaystyle -1.3 > -1\frac{1}{2}. Tess’s average lies further left, so over these rounds Tess took fewer strokes per round than Sami did, and in golf that is the better average.

(c) Better means fewer strokes, which means less. Zayn’s average is -1.1 and Yael’s is \displaystyle -\frac{4}{5}, and 1.1 > 0.8 puts -1.1 further left:

-1.1 < -\frac{4}{5}
Example 6

A lake through the year

A lake’s level is measured against its usual mark, in meters, positive above the mark. Over one year the readings were: February 0.2, April \displaystyle \frac{9}{10}, June \displaystyle -\frac{1}{4}, August -1.15, October \displaystyle -1\frac{2}{5} and December -0.45.

(a) List the months from lowest water to highest. (b) Write an inequality comparing August with October, and say what it means. (c) Write “In December the water stood lower than in June” as an inequality.

Solution

(a) Zero is the usual mark and up is positive, so lower water means a lesser number. February and April are above the mark, and 0.2 < 0.9. The four negatives in decimals are -0.25, -1.15, -1.4 and -0.45. June and December lie between -1 and 0; August and October lie between -2 and -1, so they are the two lowest. Comparing opposites, 1.4 > 1.15 and 0.45 > 0.25, so

-1\frac{2}{5} < -1.15 < -0.45 < -\frac{1}{4} < 0.2 < \frac{9}{10}

From lowest water to highest: October, August, December, June, February, April. The figure stands the number line on its end, so that higher water is higher on the page.

−1.5 −1 −0.5 0 0.5 1 October August December June February April up
Figure 4The lake of Example 6, levels in meters above and below the usual mark. Higher on the line means higher water.

(b) \displaystyle -1.15 > -1\frac{2}{5}: in August the water stood higher than in October, though both months were below the mark.

(c) December’s level is -0.45 and June’s is \displaystyle -\frac{1}{4} = -0.25. Lower means less:

-0.45 < -\frac{1}{4}

October was also the month furthest from the usual mark. That is not always so: April, \displaystyle \frac{9}{10} of a meter above, was further from the mark than June, yet April’s water was higher. Further from zero and lower agree only below the mark.

Boundaries

An exact form, not a rounded one. Rewriting keeps the point where it is only when the rewriting is exact. A decimal that never ends cannot be written in full, and rounding it can make two different numbers look equal or even swap them. When a fraction’s decimal does not stop, compare with a common denominator, or keep enough decimal places that the two numbers differ before the rounding place.

The symbols are strict. \displaystyle -\frac{1}{4} < -\frac{1}{4} is false, since no number lies to the left of itself. Two forms of the same number, such as -0.25 and \displaystyle -\frac{1}{4}, are joined by =, not by < or >.

One zero, one direction. An inequality between two measurements says something only when both are measured from the same zero with the same direction positive. A level of -0.4 meters on one lake and -0.2 meters on another, each against its own usual mark, does not say which water surface is higher; the two marks may sit at quite different heights.

A percent is compared as a number. 45\% > 0.4 compares the numbers 0.45 and 0.4. If the percents are parts of different wholes, such as 45\% of a small club and 40\% of a large school, the amounts they stand for can compare the other way.

Greater is not further from zero. On the negative side the two run in opposite directions, as Example 6 showed. Distance from zero is measured by absolute value.

Where this goes next

Putting rational numbers in order is what makes the line usable for everything after it. Finding another number between two of them is Finding a fraction between two fractions, and it relies on knowing which of the two is greater. Situations that mix balances, temperatures and heights in signed fractions and decimals are gathered in Rational numbers in real situations. Once the order is settled, the next question is how far apart two numbers are and what happens when they are combined, which is Operations with rational numbers. Inequalities with a letter standing for an unknown number, and the stretch of the number line that makes one true, grow out of the same symbols.

Exercises

Part 1: comparing two numbers

Write <, > or = between each pair.

  1. 0.43 and \displaystyle \frac{7}{16}
  2. -0.35 and \displaystyle -\frac{1}{3}
  3. \displaystyle -1\frac{3}{4} and -1.8
  4. 85\% and \displaystyle \frac{6}{7}
  5. \displaystyle -\frac{7}{8} and -0.875
  6. -2.05 and -2.1
  7. \displaystyle -\frac{5}{6} and -0.82

Part 1: ordering lists

  1. Put \displaystyle \frac{1}{2}, 52\%, 0.54 and \displaystyle \frac{6}{11} in order from least to greatest.
  2. Put -1.6, \displaystyle -1\frac{2}{3}, 0.15, \displaystyle -\frac{1}{10} and \displaystyle \frac{1}{8} in order from least to greatest.
  3. Put -3.4, \displaystyle -3\frac{3}{8}, -3.04 and \displaystyle -3\frac{2}{5} in order from greatest to least, and say which, if any, are equal.
  4. Which of -0.48, \displaystyle -\frac{3}{7}, -0.5, \displaystyle -\frac{9}{20} and -0.39 lie strictly between \displaystyle -\frac{1}{2} and -0.4?

Part 2: golf

The next season, over ten rounds, the same five golfers averaged these scores against par: Sami \displaystyle -\frac{1}{10}, Tess 1.1, Ugne \displaystyle -1\frac{9}{10}, Yael -1.7 and Zayn 0.

  1. List the golfers from best average to worst, and write the averages as one chain of inequalities.
  2. Write an inequality comparing Ugne’s average with Yael’s, and say what it means.
  3. What does Zayn’s average of 0 mean? Which golfers averaged under par?
  4. Write “Tess’s average was worse than Sami’s” as an inequality.

Part 2: the lake

The next year the lake’s readings, in meters against the usual mark, were: January -0.35, March \displaystyle \frac{3}{5}, May 1.05, July \displaystyle -\frac{1}{2}, September \displaystyle -1\frac{1}{4} and November -1.35.

  1. List the months from highest water to lowest.
  2. Write an inequality comparing September with November, and say what it means.
  3. Write “In July the water was lower than in January” as an inequality.
  4. A reading taken in one more month lay between July’s level and September’s level. Could it have been -0.95? Could it have been \displaystyle -1\frac{3}{10}?
  5. Was July’s level further from the usual mark than January’s? Was it higher?

Reasoning

  1. Explain, using a fold of the number line at 0, why of two negative numbers the one further from zero is the lesser.
  2. \displaystyle \frac{5}{7} rounded to two decimal places is 0.71. Decide which is greater, \displaystyle \frac{5}{7} or 0.71, without rounding, and explain why rounding could not decide it.
  3. Explain why every positive number is greater than every negative number, however each is written.
  4. Explain why rewriting \displaystyle \frac{3}{4} as 0.75 or as 75\% before comparing it with another number can never change the answer.
Answers
  1. \displaystyle 0.43 < \frac{7}{16}. \displaystyle \frac{7}{16} = \frac{4375}{10000} = 0.4375, and 0.4300 < 0.4375.
  2. \displaystyle -0.35 < -\frac{1}{3}. Over 300, \displaystyle 0.35 = \frac{105}{300} and \displaystyle \frac{1}{3} = \frac{100}{300}, so 0.35 is further from zero and -0.35 is further left.
  3. \displaystyle -1\frac{3}{4} > -1.8. \displaystyle 1\frac{3}{4} = 1.75 and 1.75 < 1.8, so -1.8 is further left.
  4. \displaystyle 85\% < \frac{6}{7}. Over 700, \displaystyle \frac{85}{100} = \frac{595}{700} and \displaystyle \frac{6}{7} = \frac{600}{700}.
  5. \displaystyle -\frac{7}{8} = -0.875. \displaystyle \frac{7}{8} = \frac{875}{1000} = 0.875; they are one point, so neither < nor > is true.
  6. -2.05 > -2.1. 2.05 < 2.10, so -2.1 is further from zero and further left.
  7. \displaystyle -\frac{5}{6} < -0.82. Over 300, \displaystyle \frac{5}{6} = \frac{250}{300} and \displaystyle 0.82 = \frac{246}{300}, so \displaystyle \frac{5}{6} is further from zero.
  8. \displaystyle \frac{1}{2} < 52\% < 0.54 < \frac{6}{11}. In decimals 0.50, 0.52 and 0.54; over 1100, \displaystyle 0.54 = \frac{594}{1100} and \displaystyle \frac{6}{11} = \frac{600}{1100}.
  9. \displaystyle -1\frac{2}{3} < -1.6 < -\frac{1}{10} < \frac{1}{8} < 0.15. \displaystyle 1.6 = 1\frac{3}{5} and \displaystyle \frac{2}{3} = \frac{10}{15} > \frac{9}{15} = \frac{3}{5}; \displaystyle \frac{1}{8} = 0.125 < 0.15.
  10. \displaystyle -3.04 > -3\frac{3}{8} > -3.4 = -3\frac{2}{5}. The opposites are 3.04, 3.375 and 3.4, and \displaystyle 3\frac{2}{5} = 3.4, so -3.4 and \displaystyle -3\frac{2}{5} are equal.
  11. -0.48, \displaystyle -\frac{3}{7} and \displaystyle -\frac{9}{20}. \displaystyle \frac{9}{20} = 0.45; \displaystyle \frac{3}{7} lies between \displaystyle \frac{2}{5} = \frac{14}{35} and \displaystyle \frac{1}{2}, since \displaystyle \frac{3}{7} = \frac{15}{35} and \displaystyle \frac{3}{7} = \frac{6}{14} < \frac{7}{14}. -0.5 is an end, not between, and -0.39 > -0.4.
  12. Ugne, Yael, Sami, Zayn, Tess: \displaystyle -1\frac{9}{10} < -1.7 < -\frac{1}{10} < 0 < 1.1. Fewer strokes is better, so the least average is best.
  13. \displaystyle -1\frac{9}{10} < -1.7, since 1.9 > 1.7. Ugne’s average lies further left: Ugne took fewer strokes per round than Yael, the better average.
  14. Zayn averaged exactly par. Ugne, Yael and Sami averaged under par, since their averages are less than 0.
  15. \displaystyle 1.1 > -\frac{1}{10}, or \displaystyle -\frac{1}{10} < 1.1. Worse means more strokes, which means greater.
  16. May, March, January, July, September, November: \displaystyle 1.05 > \frac{3}{5} > -0.35 > -\frac{1}{2} > -1\frac{1}{4} > -1.35.
  17. \displaystyle -1\frac{1}{4} > -1.35, since 1.25 < 1.35. The water stood higher in September than in November, though both were below the mark.
  18. \displaystyle -\frac{1}{2} < -0.35. July’s level is -0.5, further below the mark than -0.35.
  19. -0.95, yes: \displaystyle -1\frac{1}{4} < -0.95 < -\frac{1}{2}. \displaystyle -1\frac{3}{10} = -1.3, no: it is less than -1.25, below September’s level.
  20. Further from the mark, yes: \displaystyle \frac{1}{2} = 0.50 > 0.35. Higher, no: \displaystyle -\frac{1}{2} < -0.35, so July’s water was lower.
  21. For two positive numbers, the one further from zero is further right. Folding the line at 0 carries each positive number to its opposite and swaps left and right, so the partner of the number further right lands further left. That partner is the negative number further from zero, and being further left, it is the lesser.
  22. \displaystyle \frac{5}{7} > 0.71. Over 700, \displaystyle \frac{5}{7} = \frac{500}{700} and \displaystyle 0.71 = \frac{497}{700}. Rounding to two places turns \displaystyle \frac{5}{7} into 0.71 itself, so the rounded numbers are equal and cannot show which original is greater.
  23. Every negative number lies left of 0 and every positive number right of 0, so the positive one is always further right. The form only names the point; it never moves it across 0.
  24. \displaystyle \frac{3}{4}, 0.75 and 75\% all name the same point, \displaystyle \frac{75}{100}, on the line. Comparing asks which point is further right, and a new name for a point leaves it where it is.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0