How do you solve boys to girls ratios?

The ratio of the number of boys to the number of girls at school is 4:5.

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Writing a Ratio
Proportions
Practice Questions

What fraction of the students are boys?
If there are 120 boys, how many students are there altogether?

In some textbooks, a distinction is made between a ratio, which is assumed to have a common unit for both quantities, and a rate, which is defined to be a quotient of two quantities with different units (e.g. a ratio of the number of miles to the number of hours). No such distinction is made in the common core and hence, the two quantities in a ratio may or may not have a common unit. However, when there is a common unit, as in this problem, it is possible to add the two quantities and then find the ratio of each quantity with respect to the whole (often described as a part-whole relationship).

Tasks like these help build appropriate connections between ratios and fractions. Students often write ratios as fractions, but in fact we reserve fractions to represent numbers or quantities rather than relationships between quantities. For example, if we were to consider the ratio $4:5$ in this situation, then two possible ways to interpret $\frac45$ in the context are to say,

"The number of boys is $\frac45$ the number of girls,"

or to say,

"The ratio of the number of boys to the number of girls is $\frac45 : 1$."

This second interpretation reflects the fact that $\frac45$ is the unit rate (which is a number) for the ratio $4:5$.

Solution: Using a tape diagram

For every 4 boys there are 5 girls and 9 students at the school. So that means that $\frac49$ of the students are boys. $\frac49$ of the total number of students is 120 students: $$\frac49 \times ? = 120$$ If $\frac49$ the number of students is 120, then $\frac14$ of 120 is $\frac19$ of the total number of students. In other words, $\frac14 \times 120 = 30$ is $\frac19$ the total number of students. Then 9 times this amount will give the total number of students: $$9\times 30 = 270$$ So there is a total of 270 students at the school. Note that this is equivalent to finding the answer to the division problem: $$120\div \frac49 =?$$ We can see all of this very succinctly by using a tape diagram:

There are 4 units of boys and 9 units of students. Therefore 4/9 of the students are boys.
4 units = 120

1 unit = 30

9 units = 270

There are 270 students altogether.

Solution: Using a table

Boys	Girls	All students
4	5	9
40	50	90
80	100	180
120	150	270

Students can multiply the numbers in the first row by 10 to get the second row, and then double that amount to get the third row. Adding the entries in the second and third row gives the fourth row that has the solution.

Alternatively, since $120 \div 4 = 30$, students can just multiply the numbers in the first row by 30 to get the values in the fourth row.

In every row, we can see that the fraction of the students that are boys is $\frac{4}{9} $.
looking at the last row, we can see that the total number of students will be $4\times 30 + 5 \times 30 = 120 + 150 = 270$.

Page 2

A ratio compares two quantities in terms of multiplication. For example, suppose that there are 10 boys and 15 girls in a classroom. One way, not using ratios, to compare these quantities would be to say that there are 5 more girls than boys in the classroom. To employ a ratio, we say that the ratio of boys to girls is 10 to 15, or 2 to 3. In other words, for every 2 boys in the classroom, there are 3 girls.

Writing a Ratio

You can express a ratio using a colon or a fraction: namely, as

The fraction notation is usually superior for ratios with two elements, but if you have a ratio with three elements, the colon notation is usually superior: for example, 10:15:19.

In the classroom example, many other ratios can be made. The number of boys out of the total number of students is 10/25. The number of girls out of the total number of students is 15/25. And 15/10 is the ratio of the number of girls to the number of boys.

Ratios, being fractions, can sometimes be reduced. For example, the ratio of boys to girls in our hypothetical classroom is 10/15. So, we would be correct if we said that the ratio of boys to girls in our hypothetical classroom is 2:3, or two boys for every 3 girls.

Notice that knowing a ratio doesn’t tell us the actual numbers involved. If the ratio of boys to girls is 2:3 or 2/3, we don’t know whether the actual number of boys and girls is 2 and 3, 10 and 15, or 200 and 300. This fact appears frequently on GMAT questions.

Proportions

A proportion is an equation of two ratios. For example, this is a proportion:

A proportion may contain variables:

To solve such an equation, we can use cross-multiplication. In the example above, we find 3x = 60, yielding x = 20.

On many GMAT questions, you will be given a ratio, and you will solve by setting up a proportion. For example, a simple GMAT question might state, “The ratio of boys to the total number of boys and girls in a classroom is 2/3. If there are a total of 30 boys and girls in the classroom, how many boys are there?”

In this case, the question has given us a ratio, 2/3. We set up a proportion by finding something that we can set the ratio equal to:

Cross-multiplying, we find that x = 20. There are 20 boys in the classroom.