ABE Math Tutorials/Whole numbers/Cost and distance problems

Introduction
Before you work on this lesson, you should have already completed the lesson "Introduction to Word Problems". In this lesson, we'll continue with word problems:

Problems with Cost: You are probably already very familiar with the idea of "cost" and "rate" (unit cost). Have a look at the following problem:

Do you see the words $5 per kilogram? This $5 per kilogram tells you the "unit cost" of nails at this store. The little word per means "for each...". In this problem, it tells you that for each kilogram of nails, you will have to pay $5. In math, this "unit cost" is usually called a rate. It is easy to see in this problem that 2 kilograms of nails are going to cost 2 x 5 = $10. All we have to do is multiply: (number of kilograms of nails we want) times (cost of nails per kilogram [rate]) ... and we will find out the total cost for the 2 kilograms.

Objectives
Let's look at another problem: Chocolate chip cookies are on sale at the Fairview Market for 25¢ each. How much will 3 cookies cost?Can you see that 25¢ is the rate for this problem? Now all we have to do is to multiply: (number of cookies we want) times (cost of each cookie [rate]) and we can quickly see that 3 x 25 = 75¢. To find the total cost, we always multiply:

(number of items we are buying) times (cost of each item [rate]).

It is time-consuming to write this out each time, so we shorten it to just a couple of letters: (number) x (rate) = total cost Or, to make it even shorter: n x r = c Usually, we don't even bother to write in the "x". It is a general math rule, that whenever you use letters to stand for numbers, and you put two (or more) letters right next to each other, it automatically means that you must multiply them together. Now it looks like this: nr = c This tidy little mathematical sentence is called a formula. A formula is just a short-cut way of writing the kind of mathematical sentence which is always true. One good thing to note is that you don't have to memorize any formulas for the GED! All the formulas you might need will be given to you on a separate paper at the beginning of the test. This cost formula is on it, too. Practicing Problems with the Cost formula:

This table shows the sale rates on some food items:

Sale! At Fairview Fine Foods Bananas:

60¢ per kilogram Cherries: 90¢ per kilogram Onions: 75¢ per kilogram Bread: 80¢ per loaf

Using the prices in the table shown above, try answering these questions:

1. How much will 5 kilograms of bananas cost?

2. How much will 3 loaves of bread cost?

3. Susan bought 2 kilograms of cherries, 4 kilograms of onions and a loaf of bread. What was her total bill?

To check your answers, click here.

We can use the cost formula to "work backwards", too. Consider this problem: Compact discs (CD's) are on sale at Fairview Music Shop for $5 each. Tomas has $15. How many CD's can he buy?

You can probably guess the answer right away, but first let's see how this problem fits into the cost formula.In this case, we already know the total cost ( c ): we know that Tomas is going to spend the entire $15 on CD's. And we know the rate ( r ): we know that each CD costs $5. What we don't know is the number ( n ): how many CD's he can buy with a total of $15.

Let's first write the formula:

c = nr ... and now let's fit in the numbers for the letters (c and r ) that we know: 15 = n x 5 ..... and now here's the rule: when you already know the total cost and one of the other numbers (either the rate or the number), you divide to find the number that you're missing: 15 ÷ 5 =3 So we know that Tomas can buy 3 CD's at this price.

Ready for another one?

Martha was able to get 4 winter coats drycleaned for $12. What was the drycleaning cost per coat? First, we can write down the formula:

c = nr .... and now let's try to figure out which number goes where. We know that Martha got 4 coats cleaned, so "4" is the number ( n ). And we know that it cost her $12, so "12" is the total cost ( c ). The only letter we're missing is "r" -- the rate (when the question asks you to find the "cost per coat", you already know that you're looking for a rate). So when we fit these numbers into the formula, we have: 12 = 4 x r Since we already know the total cost, we know that we have another "working backwards" sort of problem just like the last one. We know that we're going to have to divide to get the answer: 12 ÷ 4 = 3 The cost per coat was $3.

Here are a few more problems to try. See if you can figure out which letter you are looking for in each problem, and then do the right operation (multiply or divide) to get the answer:

1. At the Fairview Book Store, paperback books are on sale for $3 each. How many books can Angie buy with $24?

2. At the Pet Store, puppies cost $25 each. How much will 3 puppies cost?

3. At the Bulk Bakery, Sheena spent $14 on bread. She got 7 loaves of bread for that price. What was the cost per loaf?

To check your answers, click here.

Previous Knowledge
The Distance Formula:

The distance formula is very similar to the cost formula. Have a look at this problem:

Mark is travelling at 80 kilometers per hour. How many kilometers can he travel in 2 hours? Try to picture this problem. It says that "Mark is travelling at 80 kilometers per hour". This means that in one hour, Mark will have travelled 80 kilometers. So how far will he go in two hours? It's easy to see that all we have to do is multiply:

(2 hours) x (80 kilometers per hour) = 160 kilometers ..... and the answer is that he will travel 160 kilometers in 2 hours.

We can do the same for any kind of distance problem. All we have to do is multiply the rate (kilometers per hour, meters per second, etc.) times the time (hours, minutes, etc.) and we will get the total distance travelled. Putting this into a formula, we get:

(rate) x (time) = distance ( r ) x ( t ) = d Abbreviating it even further: rt = d

Let's try another one:

Once again, we use the distance formula:

rt = d We will have to multiply the rate times the time to get the total distance that the jet will go in 4 hours. The rate is 800 kilometers per hour, so we will use "800" for r. The time, of course, is 4 hours, so we will use "4" for the t (time): 800 x 4 = 3200 So the jet will go 3200 kilometers in 4 hours.

The Distance Formula can be used to "work backwards", too. Have a look at this example:

Jason rides his mountain bicycle at a speed of 15 kilometers per hour. He travelled 60 kilometers on Saturday. How long did it take him to bike this far?

We know that we will be using the distance formula here. Which numbers do we already have? Which one are we looking for? We are asked to find "how long" it took him to travel. "How long" means that we are asked to find the time ( t ). We are also told that he travelled "60 kilometers" -- this must be his total distance ( d ). What about the "speed of 15 kilometers per hour"? That "15 kilometers per hour" sounds like a rate, doesn't it? In fact, the word "speed" is just another way to say rate in distance problems. Back to the formula, filling in the numbers that we can:

rt = d (15) x t = 60 At this point, we treat the problem the same as a "working backwards" sort of problem with the Cost Formula: when you already know the total distance, you divide to find the number that you are missing. 60 ÷ 15 = 4 So it took him 4 hours to travel that far.

Here are the steps to solving distance problems:

Figure out which number you are looking for (distance, rate, or time) Figure out which numbers you already have (rate and time, distance and time, or distance and rate) Fit the numbers that you already have, into the formula Decide whether you are going to have to multiply or divide to find the answer (multiply if you have the rate and the time; divide if you already have the distance and one other number) Do the arithmetic and state your answer.

Summary
Ready to try some distance problems on your own?
 * Joe's boat can travel at a speed of 25 kilometers per hour. At this speed, how long will it take him to travel 200 kilometers?
 * A passenger train travels at about 50 kilometers per hour. How far can this train travel in 24 hours?
 * The Vesy ferry travels from Vesy to Fairview, a distance of 34 kilometers, in 2 hours. At what speed does this ferry travel?
 * Janice can drive 240 kilometers in 3 hours. At this speed, how long will it take her to travel 400 kilometers?

To check your answers, click here. If you're ready for the homework, click here. You are probably already very familiar with the idea of "cost" and "rate" (unit cost). Have a look at the following problem:

Conclusion
At the Fairview Hardware Store, nails cost $5 per kilogram. How much will 2 kilograms of nails cost? Do you see the words $5 per kilogram? This $5 per kilogram tells you the "unit cost" of nails at this store. The little word per means "for each...". In this problem, it tells you that for each kilogram of nails, you will have to pay $5. In math, this "unit cost" is usually called a rate. It is easy to see in this problem that 2 kilograms of nails are going to cost 2 x 5 = $10. All we have to do is multiply: (number of kilograms of nails we want) times (cost of nails per kilogram [rate]) ... and we will find out the total cost for the 2 kilograms.

Let's look at another problem:

Chocolate chip cookies are on sale at the Fairview Market for 25¢ each. How much will 3 cookies cost? Can you see that 25¢ is the rate for this problem? Now all we have to do is to multiply: (number of cookies we want) times (cost of each cookie [rate]) and we can quickly see that 3 x 25 = 75¢.