Comparing Ratios: Methods, Models, and Examples

What does comparing ratios mean?

Comparing ratios means determining whether one relationship is less than, greater than, or equal to another relationship.

When two ratios share a common part, you can compare them directly by looking at the remaining values. If they do not share a common part, you must rewrite them so they share a common basis before comparing their corresponding parts.

Compare with equivalent ratios

You can compare two ratios by converting them into equivalent ratios that share the same second term.

Once the second terms match, you compare the first terms. The ratio with the larger first term is the greater ratio.

To compare ratios using equivalent forms, find a common multiple for the second terms and scale both ratios.

For example, compare the ratios $2:5$ and $3:7$.

The second terms are $5$ and $7$. Their lowest common multiple is $35$.

Scale the first ratio by multiplying both terms by $7$ to get $14:35$.

Scale the second ratio by multiplying both terms by $5$ to get $15:35$.

Because $15$ is greater than $14$, the ratio $3:7$ is greater than $2:5$.

Compare using fractions

Writing ratios as fractions allows you to use standard rules for comparing fractions.

Convert each ratio $a:b$ into the fractional form $\dfrac{a}{b}$.

Find a common denominator, rewrite the fractions, and compare the numerators.

For example, compare $5:6$ and $7:9$.

Write them as the fractions $\dfrac{5}{6}$ and $\dfrac{7}{9}$.

The lowest common denominator for $6$ and $9$ is $18$.

Multiply the numerator and denominator of $\dfrac{5}{6}$ by $3$ to get $\dfrac{15}{18}$.

Multiply the numerator and denominator of $\dfrac{7}{9}$ by $2$ to get $\dfrac{14}{18}$.

Because $\dfrac{15}{18}$ is greater than $\dfrac{14}{18}$, the ratio $5:6$ is greater than $7:9$.

Compare using a table or model

When dealing with complex contexts, organizing data into ratio tables helps track proportional relationships visually.

A table allows you to list multiples of each ratio side by side until you find a matching value in one of the columns.

Alternatively, a bar model represents the quantities as physical blocks. Models are especially useful when comparing parts to a whole or when the total amounts in both ratios share a clear relationship. By aligning the bars, you can identify the greater ratio without calculating large multiples.

Choose an efficient method

The best method for a ratio comparison depends on the given numbers.

• Use mental scaling if one second term is a simple multiple of the other, such as comparing $3:4$ and $5:8$.
• Use fractions with a common denominator when the numbers are small and share easy multiples.
• Use a unit rate, dividing the first term by the second, when finding a per-unit cost or determining speed.
• Use models when a visual representation helps clarify the relationships in a word problem.

Worked examples

Applying these methods to math word problems requires identifying the given information and selecting an appropriate strategy based on the numbers.

Example 1: Comparing ratios with the same second term

Question: Team A has a win-to-loss ratio of $5:4$. Team B has a win-to-loss ratio of $3:4$. Which team has the greater ratio of wins to losses?

Method:

1. Identify that both ratios already share the same second term, which is $4$ losses.
2. Draw a bar model for each team comparing their wins against the common $4$ losses.
3. Compare the first terms directly.

Answer: Team A has the greater ratio because $5$ wins is greater than $3$ wins for the same number of losses.

Check: Since $5 > 3$, the fraction $\dfrac{5}{4}$ is greater than $\dfrac{3}{4}$.

Example 2: Comparing by scaling fractions

Question: Recipe A uses a sugar-to-flour ratio of $2:3$. Recipe B uses a sugar-to-flour ratio of $5:6$. Which recipe has a higher ratio of sugar to flour?

Method:

1. Write the ratios as fractions: $\dfrac{2}{3}$ and $\dfrac{5}{6}$.
2. Find a common denominator. The lowest common multiple of $3$ and $6$ is $6$.
3. Scale the first fraction to match the common denominator by multiplying the numerator and denominator by $2$.
4. Compare the new numerators.

Answer: $\dfrac{2}{3}$ becomes $\dfrac{4}{6}$. Because $\dfrac{5}{6}$ is greater than $\dfrac{4}{6}$, Recipe B has the higher ratio.

Check: Converting to decimals, $2 \div 3 \approx 0.67$ and $5 \div 6 \approx 0.83$. Since $0.83 > 0.67$, the comparison holds.

Example 3: Analyzing rates with a ratio table

Question: Machine X produces $15$ parts in $2$ minutes. Machine Y produces $20$ parts in $3$ minutes. Which machine is faster?

Method:

1. Note the order of comparison: parts to minutes.
2. Set up a ratio table to list multiples of each machine's production until the minutes match.
3. For Machine X, list the multiples of the ratio $15:2$.
4. For Machine Y, list the multiples of the ratio $20:3$.
5. Compare the number of parts produced at the common time of $6$ minutes.

Answer: Machine X produces $45$ parts in $6$ minutes. Machine Y produces $40$ parts in $6$ minutes. Machine X is faster.

Check: Calculate the unit rates. Machine X produces $7.5$ parts per minute. Machine Y produces approximately $6.67$ parts per minute.

Common mistakes

Comparing ratios requires attention to proportional relationships, not just absolute values.

A common error is comparing only the first terms without ensuring the second terms match. A ratio of $7:10$ is not automatically greater than $4:5$ simply because $7$ is greater than $4$. You must establish a common basis first.

Pay attention to the order of the ratio. A ratio of $3:2$ is entirely different from a ratio of $2:3$. Reversing the order changes the meaning of the relationship and will result in an incorrect comparison.

Do not add the same value to both terms of a ratio. Adding $2$ to both parts of $3:4$ creates $5:6$, which is not an equivalent ratio. Always use multiplication or division to scale ratios.

Frequently asked questions

Can you compare ratios with different units?

No, standard ratios must compare quantities using the same units. If a problem involves different units, such as meters and centimeters, convert them to a single common unit before creating and comparing the ratios.

What is the fastest way to compare two ratios?

Converting ratios into fractions and using cross-multiplication is often the fastest method. Multiply the first term of the first ratio by the second term of the second ratio, then multiply the second term of the first ratio by the first term of the second ratio. Compare the products.

How do you compare three or more ratios?

To compare three or more ratios, write them as fractions and find a common denominator for all of them. Once all ratios are scaled to this common denominator, you can easily order them by comparing their numerators.