GED Mathematical Reasoning: Distance and Cost

In general, a formula describes a relationship between quantities. In mathematics, we use formulas to determine unknown information.

In most cases where a formula is used, all the information to solve a problem is given except for one piece of information. A formula is a tool that can be used to find that one missing piece of information. When it comes to the distance and cost formulas, 2 out of 3 pieces of information will be provided and we’ll use the appropriate formula to determine the missing piece of information.

Distance formula

The distance formula relates the quantities distance, rate and time and reads: “distance equals rate multiplied by time”. You can see that we represent “distance” with the variable “d”, rate with the variable “r”, and “time” with the variable “t”. Where distance is usually in terms of miles, rate is the same thing as speed, and time is in terms of hours.

distance = rate \times time
d = r \cdot t
d = distance
r = rate or speed
t = time

And just to review, remember that we have many ways to denote multiplication. I’ve already shown you two ways here, in conjunction with the distance formula.

We can use the multiplication sign, a centered dot, parenthesis, or – when dealing with a number and a variable or two or more variables – nothing at all. In other words, if you see a number and a variable or two or more variables right up next to each other with nothing in between, assume the operation between them is multiplication.

r \times t
r \cdot t
r(t)
r t

As a side note, some people refer to the distance formula as the “dirt” formula because it looks similar to the word “dirt.” Doing so might help you commit this important formula to memory.

Cost formula

The cost formula reads: “total cost equals number of units multiplied by the rate (or price per unit).”

In this formula, we represent “total cost” with the variable “c,” number of units with the variable “n,” and the “rate” with the variable “r”.

total cost = number of units \times rate (price per unit)
c = n r
c = total cost
n = number of units
r = rate (price per unit)

 

Example 1

How many hours will it take you to drive 175 miles if your average speed is 70 miles per hour?

 

First let’s make a note of what it is we’re being asked to find. In this case, we’re being asked to find the number of hours the trip will take.  In terms of the formula, we need to find “t.”

Next, let’s identify the information we’re given. It’s a good idea to jot this down on your paper. We’re told that the distance driven is 175 miles and the average speed, or rate, is 70 miles per hour. So we know d = 175 and r = 70.

Since we’re dealing with the three quantities of distance, rate, and speed it makes perfect sense that we would utilize the distance formula to solve this problem. When we substitute the distance of 175 and the speed or rate of 70 into the distance formula, we get

 175 = 70 \cdot t

As I mentioned previously, we will talk much more about solving equations later on, but for right now here’s the concept in a nutshell:

The statement  175 = 70 \dot t is called an equation, because it contains an equal sign stating that two things are equal to each other. To solve this equation means we will get “t” by itself on one side of the equal sign using the concept of “opposite operations”. When we solve an equation, we must do the same thing to BOTH sides to keep the equation balanced. And the answer to an equation is called the solution.

Going back to our equation, we can find the value for “t” – the time – by performing the opposite operation and dividing both sides by the number that’s with “t”, which is 70. Notice what happens when we divide both sides of this equation by 70. On the left side, we have 175 divided by 70. You may do the calculation by hand or with a calculator, which yields a result of 2.5. On the right side, 70 divided by 70 equals one, and one times “t” leaves simply “t” since anything times one is just that thing.

It is common to write the variable before the equal sign, so we can flip this around to read “t = 2.5.” It doesn’t really matter, though — the solution is the same. The time it will take to drive 175 miles going an average of 70 miles per hour is 2.5 hours.

distance1

A good final step when solving any word problem is to ask ourselves: Does the answer make sense? If so, you can feel confident that you completed the problem correctly. If not, you need to go back and examine your work for a possible mistake.

 

Example 2

If a box of cookies costs $4 and there are 16 cookies in each box, what is the cost per cookie?

 

To solve example two, we will use the cost formula, which represents the relationship between total cost, the number of units, and the rate. In this case, the rate stands for price per unit.

First let’s make a note of what it is we’re being asked to find. In this case, we’re being asked to find the price per cookie, or in other words – the rate, r.

Next, let’s identify the information we’re given. We’re told that the cost per box is $4. This is the total price. We’re also told the number of units because we’re told that there are 16 cookies in each box. So we know c = $4 and n = 16.

Since we’re dealing with the three quantities of total cost, a number of units, and a price per unit it makes sense that we would utilize the cost formula to solve this problem. When we substitute the total cost of $4 and the number of units of 16 into the cost formula, we get

c = n \cdot r \rightarrow 4 = 16 \cdot r

We can find the value for “r” – the price per cookie – by performing the opposite operation and dividing both sides by the number that’s with “r,” which is 16.distance2

Notice what happens when we divide both sides by 16. On the left side, we’ll divide 4 by 16, which we may do by hand or using a calculator.  The result is 0.25. On the right side, 16 divided by 16 equals one, and one times “r” leaves simply “r.”

You may flip the solution around to read “r = 0.25” , but the solution is the same either way it’s written. The price per cookie is $0.25.

Finally, we ask ourselves: Does the answer make sense? Yes it does. One way we can think of it is that if each cookie costs $0.25, then the cost of 4 cookies is $1. From there, it follows that 16 cookies would cost $4.

As a final note, sometimes applying the distance or cost formula is much easier than the two examples we’ve worked here. When the two values you’re given are the two quantities on the right side of the equal sign, all you need to do is multiply to find the missing quantity.

 

Example 3

Find the total cost of 4 new patio chairs costing $60 each.

 

In this situation, we’re being asked to find the total price, c. And we’re given that the number of patio chairs, n, is 4 with a price per unit, r, of $60.

Since we’re dealing with the three quantities of total cost, a number of units, and a price per unit it makes sense that we would utilize the cost formula to solve this problem. When we substitute “n” equals 4 and “r” equals 60 into the cost formula, we get

c = n \cdot r \rightarrow c = 4 \cdot 60

And so all we need to do to find the total cost, c, is multiply the right side of 4 times 60. Whether we do the multiplication by hand or using a calculator, the product is 240. So c, the total cost, equals $240.

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