Bill needed to mix up some concrete for a project he was working on in
his backyard.
He did some research and found the following table to help make a mixture
of sand and cement that would produce a concrete strong enough for his job.
So looking at the table, notice how the first row gives the number of pails of
cement, and the second row gives the corresponding number of pails of sand.
So if Bill uses three pills of cement
he needs to use seven pails of sand. If he uses six pails of cement
he must use fourteen pails of sand and so on.
And now we'll answer several questions based upon the table. A, we're asked, what is
the ratio of cement to sand? When writing a ratio
it's important that we pay attention to the order. This is asking for the ratio
of cement to sand. So going back to the table,
the ratio would be three to seven, or six to fourteen, or twelve to twenty-eight, or fifteen to thirty-five.
We can write a ratio three ways as shown here using a fraction bar, using the word
to, as well using a colon.
We typically like ratios to be simplified just like we like simplified
fractions. So let's write the ratio of cement to sand as three to seven. And for part a, we'll write
the ratio
all three ways. Three to seven using a fraction bar, three to seven using the word to, and three to
seven using a colon.
It would not be wrong to give the ratio, as let's say, twelve to twenty-eight. Notice if written
using a fraction bar, we can simplify a ratio
just like we simplify a fraction. Notice that if we divide twelve by four and twenty-eight by four, we do get
the simplified ratio of three to seven. For part b, how much sand would Bill need if
he planned to use eighteen pails of cement? Notice eighteen pails of cement is not listed
in the table. We can answer this question using a proportion. A proportion is
formed when we set two rates or ratios equal to each other.
Well we know the ratio of cement to sand is three to seven, so we have three to
seven must equal,
Bill wants to use eighteen pails of
cement and therefore, the second ratio would be eighteen pails of cement to an
unknown number of pails of sand.
We form equivalent ratios just like we form equivalent fractions. Looking at
the three and the eighteen, notice how three times six is equal to eighteen and therefore, we
multiply seven by six,
we can determine the number of pails of sand Bill needs. Seven times six is equal to
forty-two and therefore, and therefore if Phil uses eighteen pails of cement
he would need forty-two pails of sand.
For part c,
how much cement would Bill need if he used fifty-six pails of sand?
Notice here we're given the number of pails of sand. We'll be looking for
the number of pails of cement.
So the ratio of cement to sand is three to seven, so we'd have three to seven equals,
the unknown ratio is the ratio of cement to sand.
Here we're given fifty-six pails of sand and therefore, the unknown value here is a
number of pails of cement.
So we have an unknown number of pails of cement to fifty-six pails of sand. And looking at the
seven and the fifty-six. Seven times eight is equal to fifty-six,
so to find the unknown number of pails of cement, we multiply three by eight. Eight times three
equals twenty-four. If Bill uses fifty-six pails of sand, it would need twenty-four pails of
cement.
Next, how much cement would Bill need if he used twenty-eight pails of sand? Looking at the
table,
notice how twenty-eight pails of sand is given and therefore, we know Bill would need twelve pails
of cement.
Part e, what is the ratio of cement to concrete?
Notice how the number of pails of concrete is not given, but we mix the
cement and sand to get the concrete. So looking at the first column here,
if Bill uses three pails of cement and seven pails of sand
the number of pails of concrete would be three plus seven, or ten.
Let's go ahead and list the number of pails of concrete in the third row. Three plus
seven is ten. Six plus fourteen is twenty.
Twelve plus twenty-eight is forty.
Fifteen plus by thirty-five is fifty.
So going back to part e, what is the ratio of cement to concrete?
Let's go ahead and use this first row. If three pounds of cement are used, notice how
this results in ten pails of concrete. And therefore, the ratio of cement to concrete
is three to ten. And again we can write this three ways, using a fraction bar,
using the word to, or using a colon.
Again we could have used any column but using the
first column does give us the simplified ratio. If we use the ratio of
six to twenty, or twelve to forty, and simplified, we would get the ratio of three to ten.
Now for the last two questions, for part f, what is the ratio of sand to concrete? Again
let's use the first column, so the ratio is simplified. And again the ratio is
sand to concrete. The ratio of sand to concrete is seven to ten.
For part g, were asked, what percentage of concrete is sand?
Well we know the ratio of sand to concrete is seven to ten and the amount of
concrete gives us a total amount of the mixture and therefore, we can also say
that seven-tenths of the concrete is sand. So if we write the fraction
seven-tenths with the denominator of one hundred, we can determine the percent of concrete
that is sand.
Because ten times ten is equal to one hundred, and seven times ten is equal to
seventy,
we know that seventy percent of the concrete is sand.
I hope you found this helpful.