Let’s look at an example to see this in action.
Problem: Write the rational numbers $\frac{1}{3}$, $\frac{-2}{9}$, and $\frac{-4}{3}$ in ascending order.
Step 1: Find a common denominator.
We have denominators of $3$ and $9$. The easiest common denominator here is $9$.
$\frac{1}{3}$ becomes $\frac{1 \times 3}{3 \times 3} = \frac{3}{9}$
$\frac{-2}{9}$ is already at the correct denominator: $\frac{-2}{9}$
$\frac{-4}{3}$ becomes $\frac{-4 \times 3}{3 \times 3} = \frac{-12}{9}$
Step 2: Compare the numerators.
Now we look at our new numerators: $3$, $-2$, and $-12$.
Arranging these from smallest to largest, we get:
Step 3: Arrange the fractions.
Applying that order to our fractions with the common denominator:
Step 4: Final Answer (Original Form).
Now, let’s swap these back to the original numbers provided in the question:
And there you have it! By following these steps, you can confidently tackle any set of rational numbers.
Example Problem: Among $\frac{2}{-3}$, $\frac{5}{6}$, and $\frac{3}{2}$, which one is the greatest?
Put numbers in standard form: Move any negative signs to the numerator.
$\frac{-2}{3}$, $\frac{5}{6}$, $\frac{3}{2}$
Find the Least Common Multiple (L.C.M.): The L.C.M. of the denominators $3$, $6$, and $2$ is $6$.
Convert the fractions:
$\frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6}$
$\frac{5}{6}$ stays the same.
$\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}$
Compare the numerators: Since $-4 < 5 < 9$, we can clearly see the order.
$\frac{-4}{6} < \frac{5}{6} < \frac{9}{6}$, which means $\frac{-2}{3} < \frac{5}{6} < \frac{3}{2}$.
Result: The greatest number is $\frac{3}{2}$.
If you are only comparing two rational numbers ($\frac{p}{q}$ and $\frac{r}{s}$) that are in standard form, you can use cross-multiplication for a faster result.
Calculate the products of the diagonal numbers: $ps$ and $rq$.
If $ps > rq$, then $\frac{p}{q} > \frac{r}{s}$
If $ps < rq$, then $\frac{p}{q} < \frac{r}{s}$
If $ps = rq$, then $\frac{p}{q} = \frac{r}{s}$
Have you ever looked at two fractions and wondered which one is actually larger? While it’s easy to tell that $10 > 5$, it’s not always as obvious when comparing something like $\frac{4}{5}$ and $\frac{3}{7}$.
In this post, we’ll break down two foolproof methods for comparing fractions and explore the “golden rules” of inequalities when dealing with reciprocals and negative numbers.
When you need to determine which fraction is greater, you can use either the Common Denominator Method or the Cross-Multiplication Method.
This method involves making the bottom numbers (denominators) the same so you can compare the top numbers (numerators) directly.
Step 1: Find a common denominator by multiplying the denominators ($5 \times 7 = 35$).
Step 2: Convert both fractions:
$\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}$
$\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{21}{35}$
Step 3: Compare the numerators. Since $28 > 21$, we know that $\frac{4}{5} > \frac{3}{7}$.
If you’re in a hurry, cross-multiplication is much faster. Multiply the numerator of the first fraction by the denominator of the second, and vice versa.
Multiply $4 \times 7 = 28$
Multiply $3 \times 5 = 15$
Since $28 > 15$, the first fraction is greater: $\frac{4}{5} > \frac{3}{7}$.
What happens to an inequality when you “flip” the numbers into fractions (reciprocals)? It depends entirely on the signs of the numbers.
If both numbers are positive (or both are negative), flipping them reverses the inequality.
Rule: If $a < b$, then $\frac{1}{a} > \frac{1}{b}$
Example: $2 < 3 \implies \frac{1}{2} > \frac{1}{3}$
If one number is positive and the other is negative, flipping them does not change the direction of the inequality because the positive number will always stay greater than the negative one.
Rule: If $a < b$ (where $a$ is negative and $b$ is positive), then $\frac{1}{a} < \frac{1}{b}$
One of the most common mistakes in algebra happens when multiplying an inequality by a negative value.
Multiplying by a Positive ($m > 0$): The inequality stays the same.
If $a < b$, then $ma < mb$.
Multiplying by a Negative ($m < 0$): The inequality flips.
If $a < b$, then $ma > mb$.
Example:
Start with $2 < 3$. If we multiply both sides by $-1$, the relationship changes:
(Remember: on a number line, $-2$ is to the right of $-3$, making it the larger value!)