Chapter 5: Measurement

Solution

First lets express it as a rate of Miles to Gallons. We can do that using any of our formats but rates typically are written as fractions. Meaning our rate would be: $$\dfrac{300\hspace{0.25cm} \text{Miles}}{15\hspace{0.25cm}\text{Gallons}}$$

Next if we simplify that fraction down meaning divide 300 by 15 we would get our unit rate of: $$\dfrac{20 \hspace{0.25cm} \text{Miles}}{1\hspace{0.25cm} \text{Gallons}}$$

Solution

To solve this, the question is asking us to find some value $x$ that makes the fraction on the left equal the fraction on the right. To do that we use cross multiplication and solve for $x$.

\begin{align*} \dfrac{5}{3}&=\dfrac{x}{6}\\ \\ 5(6) &= 3 (x) \\ \\ 30 &= 3x \\ \\ \dfrac{30}{{\color{red}3}} &= \dfrac{3x}{{\color{red}3}}\\ \\ 10 &= x\\ \end{align*}

Thus we can see that $\dfrac{5}{3}=\dfrac{10}{6}$

Solution

We can find the answer to this by setting up our proportion where we compare inches to miles, and allowing the variable $x$ to represent the missing measurement we don't know.

$$\dfrac{\frac{1}{2}\hspace{0.25cm}\text{inches}}{3\hspace{0.25cm}\text{miles}}=\dfrac{2\frac{1}{2}\hspace{0.25cm}\text{inches}}{x\hspace{0.25cm}\text{miles}}$$

Once again we start by cross multiplying to get: $$\left(\frac{1}{2}\right)x=3\left(2\frac{1}{4}\right)$$

In order to do this we will need to convert the mixed number of $\left(2\frac{1}{4}\right)$ to an improper fraction. As a refresher this can be done by multiplying the whole number times the denominator $2(4)$ then adding the numerator $1$ and putting that number over the denominator to get $\dfrac{2(4)+1}{4}=\dfrac{9}{4}$. Put that back into our equation now:

$$\left(\frac{1}{2}\right)x=3\left(\dfrac{9}{4}\right)$$

Remember when multiply fractions to whole numbers we assume the whole number is a fraction with a denominator of one and then we multiply numerator to numerator and denominator to denominator.

$$\left(\frac{1}{2}\right)x=\dfrac{3}{1}\left(\dfrac{9}{4}\right)$$ $$\left(\frac{1}{2}\right)x=\dfrac{27}{4}$$

Next to finish this problem up we would divide by $\frac{1}{2}$ on both sides to get $x$ alone. Recall that dividing by a fraction is the same as multiplying be its reciprocal (flip the fraction).

$$\left({\color{red}\frac{2}{1}}\right)\left(\frac{1}{2}\right)x=\dfrac{27}{4}\left({\color{red}\frac{2}{1}}\right)$$ $$x=\dfrac{54}{4}$$ $$x=\dfrac{27}{2}$$ $$x=13\dfrac{1}{2}$$

Calculator Tips

Fractions are hard! Do not get caught up in the fractions you have a calculator that will convert numbers to and from. If you are using a TI-30 family calculator you should have a PRB key, this key when you hit the second key 2nd first will convert a decimal to a fraction or a fraction to a decimal you should see $F\blacktriangleleft\blacktriangleright D$ above the key PRB.

Additionally you should have a A $^b/c$ below the PRB. This key allows you to input mixed numbers. If you wanted to input $\left(2\frac{1}{4}\right)$ into the calculator you can do so by entering 2 A $^b/c$ 1 A $^b/c$ 4 on the screen it will look like $2 \lrcorner 1\lrcorner 4$, then hit the ENTER and you should see your mixed number.

Lastly above the A $^b/c$ you should see $A ^{b}/{c}\blacktriangleleft\blacktriangleright \frac{d}{e}$ above your A $^b/c$. This key allows you convert between a mixed number and an improper fraction. For example is we entered our answer $\dfrac{27}{2}$ into our calculator so 2 7 / 2 ENTER followed by 2nd PRB ENTER it should convert your fraction to a mixed number. Once you get familiar with these keys your life will be a lot easier!