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Cost and distance problems

Whole numbers

Introduction | Place value | Rounding | Long addition & subtraction | Long Multiplication | Long division | Expressing operations | Word problems | Order of operations | "Set-up" problems | Cost and distance problems | Introduction to algebra | Powers of 10 | Estimation | Not enough info | Homework

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:

Activity

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.

Tomatoes were selling for $1.20 per kilogram.Apples were selling for $1.00 per kilogram.Acustomer paid 42.40 for tomatoes and $2.00 for apples.How can you find out how much 4 kilograms of tomatoes cost?.

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:

Activity

A fast jet can travel 800 kilometers per hour. How many kilometers can it go in 4 hours?

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¢.