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Ratio of boys to girls


Alignments to Content Standards: 6.RP.A

Task

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

  1. What fraction of the students are boys?

  2. If there are 120 boys, how many students are there altogether?

IM Commentary

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 $4:5$.

Solutions

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:

Bar_diagram_67722d3e6005449e4fbe5c57eaad4f3c

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

  2. 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.

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

Oluwatosin says:

about 3 years

The ratio of the boys to girls is 4:5, which implies that the number of boys and girls are 4a and 5a where a represents a positive integer because number of people cannot be fraction or negative. From this step, I was able to find the total number of students in terms of a by solving 4a + 5a = 9a. To find the fraction of the students that are boys, I divide the number of boys in the school with the total number of students by doing 4a/9a which gives 4/9.

To find the total number of students given the number of boys are 120 in all, I will use the number of boys 4a to find the value of the variable a 4a = 120; Dividing both sides by 4 gives a = 30. Because we have established that the total number of students = 9a, we can then find the total number of students using a = 30 in our expression for total number of students. This implies that total number of students = 9a = 9 (30) = 270 students.

Kristin says:

about 6 years

I've added the alternative solution suggested by Bridget. Continued suggestions welcome!

Bridget Dunbar says:

about 6 years

I believe that ratio tables are essential for building proportional reasoning at this age. Part b of this task can easily be represented using a ratio table.

I'm having a hard time pasting a table into this comment box...however, the two labels would be number of boys and number of girls. The first entry into the ratio table would be 4 boys and 5 girls. In order to scale up to 120 boys...students would need to multiply 4 times 30. Therefore, 5 times 30 is 150. Hence, finding the sum of the boys and girls would yield 270 students.

Robert says:

over 4 years

I agree with Bridget Dunbar, but feel a simpler solution would be to treat the ratios as equivalent fractions. 4/9 = 120/n. Since 4 x 30 = 120, 9 x 30 = 270. Therefore, 4/9 = 120/270 is a valid proportion. For 120 boys, there must be 270 students.

Kristin says:

about 6 years

Thanks for your suggestion for an alternative solution! I'll work on this as soon as I can.

agrannemann says:

over 6 years

I like the task and dislike the solution. Here is a problem that forces the agenda. Part a is a GREAT question. Solving part b from part a is cumbersome. You just lost half the class, unless you are willing to use a proportion and represent the unknown with x. Proportions are the best approach in this case.

EllenM says:

almost 6 years

I like the first and second solution. I tend to think along the lines of the first solution in my head -- since 4/9 of the students are boys, and that quantity is 120, I divide the 120 boys by 4 to figure out how many students make up 1/9 of the total, and I get 30, which represents one small rectangle from the tape diagram. Then, since the total number of students is 9/9, I just multiply the 30 (which is 1/9 of the total) by 9 to get the total number of students, and it is clear from the diagram that there are 9 groups of 30 making up the total. I understand that this may feel cumbersome if you don't think that way, but it makes perfect sense to me. I think many students learn the more efficient way of setting up a proportion and solving for x, but they may not fully understand what they are doing and why, and this may lead to problems when the questions are not so straightforward.