# Composing expressions

In the previous section we used an example where we wanted to know how many hours a person works over a period of two days if he each day were to work 4 hours before lunch and 3 hours after lunch. As we could see this problem could be turned into the expression

$$\left ( 4+3 \right )\cdot 2=14$$

When we want to translate a verbal phrase like the example above into a mathematical expression we can look for words that indicate a mathematical operation. For example words like "sum", "increased by" and "plus" indicates that we are to use addition. And words like "times" and "multiplied by" calls for multiplication. When we are writing subtractions and divisions the order in which we write is important. "The difference of a number 12 and a" is written 12 - a and not a - 12.

When we have a mathematical problem as in the example below we can begin by making a verbal model where we describe the situation in words and relate the words by usage of mathematical symbols. These words can then be replaced by numbers or variables to create a mathematical model, or expression, to describe the situation.

**Example**

Tony is at a car rental service to rent a car. There is an administration fee of $50 to rent a car and then it costs $20 for each day he has the car. Write an expression for the total cost of renting a car at this particular car rental service.

Written as a verbal model

$$administration\: fee+price \: each \: day\cdot number\: of\: days$$

Translate the verbal model into a mathematical model

$$50+20x$$

where x are the number of days that you've rented the car.

When we measure how fast a car or something is moving we are usually comparing quantities measured in different units like comparing distances with time. The fraction resulting is called a rate. The unit rate is when the denominator of the fraction is 1 unit.

**Example**

An airplane travels 760 miles in 2 hours. Calculate the unit rate.

$$\frac{760\: miles}{2\: hours}=\frac{760\: miles\div2 }{2\: hours\div 2}=\frac{380\: miles}{1\: hour}$$

If you're uncertain as to whether or not your expression is resulting in the wanted unit you can always do a unit analysis. In a unit analysis you exchange the numbers and variables in the expression with the corresponding units. Here we use the verbal model from example 1.

$$\begin{matrix}administration\: fee \\ \left ( dollars \right ) \end{matrix}+\begin{matrix}price \: each \: day \\ \left ( dollars\div days \right ) \end{matrix}\cdot\begin{matrix}number\: of\: days \\ \left ( days \right ) \end{matrix}$$

And to test what the resulting unit is we only keep the units in the expression:

$$dollars+\frac{dollars}{day} \cdot days=dollars+dollars=dollars$$

**Video lesson**

Write an expression that describes the situation:

Anna and her parents are going to the movies. Each ticket costs $8 and Anna also wants popcorn which is an additional $3. How much did they pay.