How To Solve Ratios Using Ratio Strategies

Many ratio problems can be solved efficiently without ever using cross-multiplication. These strategies engage students in thinking about relationships in ratios and comparing the given values in different ways. Thinking about equivalent ratios should be a problem-solving strategy that students can apply to many different types of problems.

How To Solve Ratios By Simplifying?

How many times have you watched students use cross multiplication to find the missing value in a ratio problem such as 18/15=x/20. Cross multiplication will certainly work, but 18 times 20 divided by 15 is not a simple computation. More importantly, while struggling over the computation for 18 times 20 divided by 15 a student is probably thinking very little about how the two ratios are related or what kind of answer will make sense.

An initial step that will serve students well on nearly any ratio problem is to determine whether the initial ratio can be simplified. The given ratio of 18/15 can be simplified since 18 and 15 both contain a factor of 3.

 18/15 div 3/3=6/5

Now we have 6/5=x/20  and the relationship between the two ratios is more clear.

Since 5 x 4 = 20, we can multiply 6 x 4 to find that x = 24.

Row Multiplier

Tables are frequently used to present equivalent ratios. Students are generally asked to find one or more missing numbers. The relationships between the values that are given can be used to find missing values.

The row multiplier strategy means looking for a relationship between the numbers in different rows. Then use that relationship to fill in the missing values.

Since we multiply by 2 to get from the first row to the second row for the values on the left side of the table, we need to use that same multiplier for the values on the right.

Since we multiply by 3 to get from the first row to the third row for the values on the right, we need to use that same multiplier for the values on the left.

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The more rows that are filled in on the table, the more opportunities there are to look for relationships between rows. If this table had an additional row, we could use any of the numbers on the right to find the missing multiplier. 

Any multiplier will result in a value on the left of 36. Students thinking about the relationships between the rows are engaged with the relationships in the table and making choices about how to solve them. 

Column Multiplier

This strategy is similar to the row multiplier strategy, but we are looking for a relationship between columns.

In the given ratio we can see the number on the left is multiplied by 2 to get the number on the right. We follow that same rule to fill in the missing value in the next row.

For the next row, we are moving from the right column to the left. This is a great opportunity to think about inverse operations. Some students will recognize that we need to divide instead of multiplying. It’s also an opportunity to reinforce the relationship between division and multiplication by the reciprocal. Some students will suggest multiplying by ½. Either strategy will work, and both will build stronger math students than putting the numbers into a proportion and using cross multiplication to solve.

Finding a Common Factor

Part one included a strategy of simplifying a ratio as a first step. This strategy is about simplifying in a way that will allow students to get to the needed ratio more quickly.

18/42=27/x

We could simplify this ratio by dividing both numbers by the common factor of 6, resulting in 3/7.  Then we could multiply both parts of the ratio by 9 and see that x = 63. Another option is to think about the relationship between the numbers in the given ratio and the numbers in the equivalent ratio we are trying to achieve.

Both given numbers are even. If we halved each one we would have a value of 9 on top, which can be multiplied by 3 to reach 27. 

Finding a common factor isn’t necessarily a shortcut or easier than changing the original ratio to the simplest form and multiplying, but it does show a more solid understanding of how ratios can be manipulated.

Adding or Subtracting Rows

This strategy is based on the understanding that every row in a ratio table is equivalent. Each row looks slightly different, but the same comparison of numbers is always represented. 

Notice in this table the sum of the first two rows in the column on the left is equal to the third row. The same pattern is true for the column on the right.

To find the final missing value, we could add the values for rows 3 and 4 to get 105. Think of this as taking three groups of objects, each in the ratio 1:15 and combining them with four groups of objects, each in the ratio 1:15 . The missing value for the final equivalent ratio is 105.

This understanding provides another tool for finding missing values for equivalent ratios.  In the problem, 40/48=35/x we see that both numbers share a common factor of 8, and when simplified the ratio becomes 56. The ratio 40/48 is essentially 8 groups of objects in the ratio 56. We are trying to reach a value of 35 in the equivalent ratio, and 35 is 40 – 5. In other words, we remove one group of objects from the given ratio. 40/48-5/6=35/42. So x = 42. 

We could use a similar strategy to solve 40/48=45/x. Instead of subtracting a group, we would add a group. 40/48+5/6=45/54.

Also read: How to Divide Fractions into 3 Easy Steps With Examples

Takeaway 

Ratio strategies help students move beyond memorizing the steps for cross-multiplication toward a sense of having many tools at their disposal. They allow students to take charge of their problem-solving in order to demonstrate a deeper understanding of ratio concepts.

Frequently Asked Questions on How To Solve Ratios?