Welcome to a lesson on unit conversions. In this lesson we will focus on non-
standard units of measure. We will perform the conversions using unit fractions. In
the first example, farmer Tom has a barter system, where four pigs equals nine
ducks, one goat equals three ducks and two pigs equals five chickens.
The first question is for part a. Farmer Tom needs to figure out how many chickens
he can trade for six goats. Another way to write this is six goats equals how
many chickens?
To begin we'll write six goats as a fraction with a denominator of one. From
here we multiply by unit fractions to convert goats to chickens.
We form the unit fractions using the conversions shown here. Because one goat equals three
ducks, we can use this conversion to find our unit fraction to convert goats to ducks.
Because we want goats to simplify out, and right now we have goats in the numerator.
For the unit fraction we need goats in the denominator, so that goats simplify out.
Because the conversion is one goat equals three ducks,
we have the unit of ducks in the numerator and the conversion is three
ducks equals one goat.
Because these are equal, this quotient is equal to one. So multiplying by this
unit fraction does not change the value, it just changes the units. Notice goats divided
by goats simplifies to one.
But our goal is to have chickens, not ducks, so now we'll multiply by another
unit fraction using the conversion four pigs equals nine ducks.
Because we want ducks to simplify out, we must have ducks in the denominator.
Because ducks are in the numerator here.
The conversion is four pigs equals nine ducks, so we'll have pigs in the numerator and
the conversion is four pigs equals nine ducks.
The units of ducks simplify out, or simplify to one.
Now we have pigs, but we still want chickens.
So we use our final conversion to multiply by one more unit fraction using the conversion:
two pigs equals five chickens.
We want pigs to simplify out, so we have pigs in the denominator and chickens in the
numerator. The conversion is two pigs equals five chickens, so the unit fraction
is five chickens divided by two pigs. The units of pigs simplify out, or simplify to one.
Notice now we are left with the units of chickens, which is what we want. So now we
multiply the fractions. We could simplify before multiplying.
We'll let's go ahead and multiply the numerators and multiply the denominators.
So six times three, times four, times five equals three hundred sixty.
And one times one, times nine, times two is equal to eighteen. Three hundred sixty divided by eighteen
is equal to twenty.
So now we know that six goats equals twenty chickens.
Let's look at some more examples.
Here we have the same barter system with some additional conversions. For a, how
many goats can be traded for eighteen geese?
So we begin with eighteen geese as a fraction with the denominator of one.
We want to convert geese to goats. So on the far right we have a certain
number of goats,
which we will determine. Now we need to figure out which conversions to use.
We begin with geese, and notice how here's a conversion that says five ducks
equals six geese.
And here's another that says one cow equals twelve geese. If we convert to cows
though, there are no other conversions that we can use and therefore, we'll use
the conversion five ducks equals six geese to begin. Because we want the units of geese
to simplify out, we multiply by a unit fraction where we have geese in the denominator and ducks
in the numerator. The conversion is five ducks equals six geese, so the unit
fraction is five ducks divided by six geese. Units of geese simplify out leaving us with
ducks. Now we look for a conversion involving ducks that we can hopefully use to
convert to goats.
Notice how here, we're given one goat equals three ducks and therefore, we can
multiply by one more unit fraction to convert ducks to goats. We want ducks to
simplify out, so we must have ducks in the denominator and we'll have goats the
numerator. The conversion is one goat equals three ducks and therefore, the
unit fraction is one goat divided by three ducks. The units of ducks simplifty to one. And now
we multiply the fractions. Again we could simplify before multiplying,
but let's just go ahead and multiply. In the numerator we have eighteen times five, which
equals ninety.
In the denominator we have six times three, which equals eighteen. Ninety divided by eighteen is
equal to five and therefore, eight geese equals five goats.
To answer the question, how many goats can be traded for eighteen geese?
We now know the answer is five goats.
For part b, if you need ten horses,
how many pigeons would you need to barter? To answer this question we'll
convert ten horses to pigeons.
Looking at the conversions,
we can use the conversion, five horses equals three bulls to convert horses to bulls.
And then we can use the conversion one bull equals thirty-two pigeons to convert bulls
to pigeons.
So for the first unit fraction,
because we want horses to simplify out, we need horses in the denominator.
And because the conversion is five horses equals three bulls, we have bulls in the
numerator and the unit fraction is three bulls divided by five horses. Notice now the units
of horses simplify out, or simplify to one. And now we'll use a conversion: one bull
equals thirty-two pigeons to convert bulls to pigeons and answer our question. We
want bulls to simplify out, so bulls must be in the denominator, and we have
pigeons in the numerator. The conversion is one bull equals thirty-two pigeons and
therefore, the unit fraction is thirty-two pigeons divided by one bull. The units of bulls simplify to one.
And now we multiply the fractions, which will tell us how many pigeons we need to
barter for ten horses. Multiplying the numerators,
we have ten times three, times thirty-two, which equals nine hundred sixty.
In the denominator we have one times five times
one, which equals five. Nine hundred sixty divided by five is equal to one hundred ninety-two.
So now we know ten horses equals one hundred ninety-two pigeons.
Going back to the question, if you need ten horses, how many pigeons
would you need to barter? And the answer is one hundred ninety-two pigeons.
I hope you found this helpful.