The ratio of girls
to all students is five to eight.
Select whether each statement is true or false.
Knowing the ratio of girls to all students
does not indicate how many girls,
or how many students, are in the classroom,
but what it does tell us is if we were
to take all of the students,
and put them into eight groups of equal size,
with each group only having girls or only having boys,
five of the eight groups would be girls.
And we can model this ratio using the bar tape diagram
shown here below, where the entire bar
represents all of the students,
and because the ratio of girls to all students
is five to eight, we cut the bar
into eight equally sized parts,
each part represents a group of students,
and then shade five of the groups,
because the ratio of girls to all students is five to eight.
Because all students are girls or boys,
we also know that these three parts,
or these three groups, must be boys.
Before we address the four statements,
let's list all of the known ratios from the model,
and let's use a fraction bar for the ratios,
rather than the colon.
So, the given ratio of girls to all students
is five to eight, the ratio of boys to all students
is three to eight.
The ratio of girls to boys is five to three,
and the ratio of boys to girls is three to five.
And now, let's determine whether the statements
are true or false, for a, there could be
nine boys to 15 girls in the classroom.
Well, the known ratio for boys to girls is three to five.
To determine whether the first statement is true or false,
we need to determine whether the ratio of three to five
is equivalent to the ratio of nine to 15.
We determine whether the ratios are equivalent
just like we determine whether fractions are equivalent.
If we multiply the three and the five by the same value,
to get the ratio on the right side of the equals sign,
then the two ratios are equivalent.
Notice three times three is equal to nine,
and five times three is equal to 15,
showing these two ratios are equivalent.
Statement a is true.
We can model the ratio of boys to girls as nine to 15,
if each group has three students.
Notice the ratio of boys to girls is nine to 15.
For statement b, there could be
10 girls to 16 boys in the classroom.
The known ratio of girls to boys is five to three.
We need to determine whether the ratio of five to three
is equivalent to the ratio of 10 to 16.
Five times two is equal to 10, but notice,
three times two is equal to six, not 16,
which means these two ratios are not equivalent,
and therefore, statement b is false.
Statement c states, there could be
20 girls to 32 students in the classroom.
The ratio of girls to all students was given
as five to eight, we need to determine
whether the ratio of five to eight
is equivalent to the ratio of 20 to 32.
Starting with the five, notice five times 4 is equal to 20,
and eight times four is equal to 32.
These two ratios are equivalent, statement c is true.
To model the ratio of 20 girls to 32 students,
each group of students would need to have four students.
Notice this model does show the ratio
of girls to all students as 20 to 32.
And for statement d, there could be
12 boys to 36 students in the classroom.
The ratio of boys to all students must be three to eight.
We need to determine whether the ratio of three to eight
is equivalent to the ratio of 12 to 36.
Starting with the three, notice three times four
is equal to 12, but eight times four is equal to 32, not 36.
Which means, these two ratios are not equivalent,
and therefore, statement d is false.
I hope you found this helpful.