Problem Solving with Rates & Patterns


Algebra is the part of mathematics in which letters and other general symbols are used as a language to represent numbers and quantities in formula and equations. Examples of Algebra can be as simple as “how many burgers do you need to make for five people, if everyone will eat two burgers” to advanced scientific mechanics to designing your smartphone. Algebra is the formula we use to build and discover everything around us.

For the following rate and ratio questions, use multiplication and division. As always, show your work and make sure you consistently include your units.

Question 1.1

If there are 6 apples per packet, how many apples do we have if we have 3 packets of apples?

What do we know from the question? There are 6 apples PER packet. For these types of questions, it helps to think of that phrase as “6 apples per 1 packet”. A reminder, “per” also can be thought of as division or a “ / “

We want to know how many apples are in 3 packets, so we will use multiplication.

3 packets
x
6 apples
1 packet
=  18 apples

Note how the "packets" cancel each other out

Question 1.2

If a bus is travelling at 45 miles per hour, how far will it travel in 4 hours?

Again we use multiplication. The bus is travelling at 45 miles per 1 hour or ((45 Miles)/(1 Hour))

So,

4 hours
x
45 miles
1 hour
=  180 miles

Note again how the "hours" cancel each other out

Question 1.3

A pumpkin pie has 12 slices and 4 people want to split it evenly, how many slices are there per person?

This one we will have to use division. A hint to knowing this is in the question, it wants to know how many slices are there per person or slices/person.

We have 12 slices and we want to split (or divide) it amongst 4 people.

12 slices
4 people
=  3 slices/ person (or 3 slices per person)

Question 1.4

If a runner travels 10 miles in one hour, how many minutes does its take per mile travelled?

We know, one hour = 60 minutes, and we want to know minutes per mile travelled or minutes/mile.

60 minutes
10 miles
=  6 minutes per mile
60 minutes
10 miles
can also be written as 60 minutes / 10 miles or 60 minutes / 10 miles.

Question 2.1

A tap pours out 60 milliliters (or 60 ml) per second.

Create an equation that gives the volume, V (ml) , the tap has poured where the variable that you know is time, t (seconds).

For this equation we do not need to include units as the symbols in the question have been described with units already.

An equation that GIVES the volume, V. That means and equation for V or an equation that starts like

V =

We know that after one second we have the volume = 60 x 1 after two seconds the volume = 60 x 2 and so on. So for the equation we write

V = 60t

Question 2.1a

How long does it take for the hose to pour a volume of 900 ml?

For this question, we know the volume, V, that we want and its now asking us for the time, t.

We use the equation from a)

V = 60t

Then substitute what we have been told from the question (V=900)

900 = 60t

And then start to re-arrange to have an equation that gives us the value of t.

Divide both sides by 60

900
60
=
60t
60
15 = t

We will now get more in to the fundamentals of Algebra as we start to increase the complexity of the relationship between sentences and words and symbolise that in equations.

Question 2.2

A motorbike and a car are driving towards each other down a single track road that is 150 miles long. They each start at each end of the road, 150 miles a part. The motorbike is travelling at 3 miles per minute and the car is travelling at 2 miles per minute.

What distance, d, will the motorbike and car have travelled each in miles after:

i) 1 minute

ii) 8 minutes

iii) 15 minutes

iv) 20 minutes

v) 20 minutes in total together

We need to multiply the speed (in miles per minute) by the number of minutes. So:

i)After 1 minute: Motorbike's distance = 3 x 1 = 3 and car's distance = 2 x 1 = 2

ii) 8 minutes: Motorbike's distance = 3 x 8 = 24 and car's distance = 2 x 8 = 16

iii) 15 minutes: Motorbike's distance = 3 x 15 = 45 and car's distance = 2 x 15 = 30

iv) 20 minutes: Motorbike's distance = 3 x 20 = 60 and car's distance = 2 x 20 = 40

v) What is the total distance of the car and motorbike after 20 minutes?

Simply add the distance of the car and motorbike from iv)

v) 60 + 40 = 100

Let us develop these questions further:

Create equations for each distance in miles, d, that the motorbike and car have travelled over time in minutes, t and use them to calculate at what time they will meet in the middle.

First, find equations that gives us the value of distance, d, depending on the time passed, t.

Note: This is similar to question 2.1.

Truck: d = 3t

Car: d = 2t

Secondly, what must be true about the total distance travelled by the motorbike and car for them to meet in the middle?

Try thinking of the question as, “What must be true about the total distance travelled by the motorbike and car for them to meet in the middle of the 150 mile long road?”

If they have each started at each end of the road, 150 miles apart and meet at some point in the middle, the total they have both travelled can't be any more or any less than 150 miles.

The question isn't looking for how far the car and motorbike have travelled individually, it just needs to know the total distance travelled.

Another way of thinking about it would be, imagine the car and motorbike have met in the middle, and you knew the distance that the car had travelled and the motorbike had travelled, if you added the two distances together, what would it be? Yes - it would be 150 miles. We know that when the car and motorbike meet in the middle the total distance travelled is equal to 150 miles.

How do we find out the total distance travelled by the car and motorbike? Add the two distances together. What's another way we can represent the distance the car and motorbike travel individually?

Truck, d = 3t and car, d = 2t.

Adding these two equations for distances together will give us

2t + 3t = total distance travelled (which we know also) = 150

so

2t + 3t = 150

5t = 150

5t
5
=
150
5

t = 30

So we know that the car and motorbike will meet at t = 30, or 30 minutes.

Fluency and Language

In particular, at this stage, fluency means understanding what is being asked and identifying whether to use multiplication or division.

Question 3

Tim is running at 120 steps per minute. If Tim runs for 6 minutes, how many steps will he have taken?

Start with writing what we know from the question

Tim is running at 120 steps per minute, remember “per” also means divide, and it also helps to write minute as 1 minute, so we can write that statement as

120steps / 1 minute

So in one minute, Tim runs 120 steps. The question asks us how many steps Tim takes in 6 minutes, think of as Tim running 1 minute, 6 times. 6 times one minute. We now start to see that we are going to use MULTIPLICATION.

Writing out our work will look like:

Tim runs 120 steps / 1 minute for 6 minutes

Or:

 120 steps
1 minute
x 6 minutes
Which is the same as
120 steps x 6 minutes
1 minute

With the minutes cancelling out (remember, anything divided by itself is equal to 1) we are left with,

120 steps x 6 = 720 steps

If you write down what you know from the question including all the units, and cancel out units through your working, you will always be left with the right unit.

Extra tip:

Another way to figure out if we need to use multiplication or division is by looking ahead at the unit that the ANSWER must be in. Our answer is going to be a certain amount of steps, say that the number of steps we want is represented by the letter x.

So are answer will look like,

x steps.

From what the questions gives us, we know we will be working with. i.e. 120 steps per minute or 120 steps / 1 minute and also 6 minutes.

Think about what we will have to do with this to end up with an answer that just has steps as the units. The minutes will have to cancel out, so we will have to multiply.

Question 4

David bought 6 bags of candy. Each bag contains 35 pieces of candy per bag. How many pieces of candy in total does he have.

Start with writing what the question has given us:

David has

6 bags

And each bag has

35 pieces per 1 bag, or 35 pieces / 1 bag or 35 pieces / bag

He has 1 bag, 6 times.

1 bag is 35 pieces per 1 bag, 35pieces / 1 bag

Times that by 6 bags, will look like

(35 pieces / 1 bag) x 6 bags =

35 pieces x 6 bags
1 bag
= 6 x 35 pieces = 210 pieces.

Remember, the unit of our answer will have to be in pieces. The answer gives us 35 pieces/bag and 6 bags, to have the unit pieces remaining, the bags must cancel out.

Question 5

Sam has 28 treats that he wants to divide evenly to his 4 cats, he distributes the treats evenly amongst 4 bowls, how many treats are there per bowl?

Write what we know from the question.

There are 28 treats.

There are 4 bowls.

The question says Sam wants to DIVIDE the treats evenly, which gives us a hint that we will use division.

We can also use the extra Tip regarding units. We want the answer in treats per bowl or treats / bowl. The question gives us 28 treats and 4 bowls, to get the unit of treats per bowl (treats/ bowl) we will have to divide the amount of treats by the amount of bowls.

28 treats
4 bowls
= 7 treats / bowl or 7 treats per bowl

Question 6

Oleg is selling paintings for $100 each. If he has sold 7 paintings, how much money has he made?

Write what we know from the question.

This time the units given are not as obvious.

Oleg is selling paintings for $100 each it may help to rewrite this as:

$100 per painting

or

$100/ painting

(If it helps, you can also write it as 100 dollars / painting as long as you stay consistent with that unit and then change the “dollars” back to $ at the end, for now we will stick with the $.)

The question also gives us that Oleg sold,

7 paintings

The questions asks us how much he makes from selling 7 paintings, so the unit the answer will be in will be $ or dollars. Again, the units we want are not as obvious from the question.

Oleg sold 7 paintings at $100 / painting

So,

$100
Painting
x 7 Paintings = $700

Oleg made $700

Question 7

If there are 6 buildings in a block, and 10 apartments in a building, and 4 rooms in an apartment, then how many rooms are in a block?

This question is worded slightly differently, but we still use the same method we've been using. This is where we learn to change the language we speak into the language of algebra.

Write what we know from the question

6 buildings in a block

10 apartments in a building

4 rooms in an apartment

And we want to know how many rooms are in a block.

If we first start by changing “in a” to “per” and “per to “/” that will help visualize the math that we have to do.

6 buildings in a block -> 6 buildings per 1 block -> 6 buildings / 1 block

10 apartments in a building -> 10 apartments per 1 building -> 10 apartments / 1 building

4 rooms in an apartment -> 4rooms per 1 apartment -> 4 rooms / 1 apartment

We want to know how many rooms are in a block, so rewrite as:

-> rooms per 1 block -> rooms / 1 block

First, work out how many rooms are in a building

4 rooms per apartment and 10 apartments per building,

So one building has 10 apartments with 4 rooms per apartment.

10 apartments
building
x
4 rooms
apartment
=

(apartment/ apartment cancelling out)

(10 x 4rooms) / building

40 rooms / building

40 rooms per building,

40 rooms in a building.

Now, use the same method for amount of rooms per block

6 buildings / block and 40 rooms / building,

6 buildings
block
x
40 rooms
building
=

(buildings/ building cancelling out)

= (6 x 40 rooms) / block =

240 rooms / block or,

240 rooms per block or,

240 rooms in a block

If we have a strong understanding of this and once we are familiar with the methods, we can do it all at once

(6 buildings / block) x (10 apartments / building) x (4 rooms / apartment) =

6 buildings x 10 apartments x 4 rooms
block x building x apartment
=

With buildings x apartments on top cancelling out with building x apartment below.

(6 x 10 x 4 x rooms) / block =

240 rooms / block

Also, if we use our extra Tip method, we will have to approach this one differently. Looking at the units of the answer we want, we have rooms / block, so we will want rooms as the numerator and block as the denominator.

Just because the unit of the answer is rooms/block does not necessarily mean we will use division. Looking at what the question gives us we have the units buildings/block, apartments/building, and rooms/apartment

Notice, we already have here rooms as the numerator and blocks as the dominator, and if we were to use division, it would switch this around. Notice also that if all of those units were multiplied together, apartments and buildings would be cancelled out, leaving us with rooms/ block. This hinting indicates the method that we should use.

Another way to help is to use a visual:

apartment block

Question 8

Tim is running at a speed of 9 miles per hour. In 2.5 hours how far will he have run?

Write what we know from the question:

Tim running at 9 miles per hour or, 9 miles / hour.

Tim runs for 2.5 hours. So,

For 2.5 hours, Tim runs at 9 miles / hour

2.5 hours
x
9 miles
hour

(hours/hour cancels out)

2.5 x 9 miles = 22.5 miles

Question 9

Mr. Lahey is a teacher who wants to know how many students he will have at each of his classroom tables in his class of 35 students. He has 7 tables. How many students will be at each table, evenly distributed?

Write what we know,

35 students

7 tables

We want to know how many students will be at each table, again this is where we learn to turn our spoken language into the language of algebra. Students at each table means students per table, which means students/table. So divide:

35 students / 7 tables =

5 students / table or 5 students per table.

The following 5 questions related to the block of apartments build above a store as illustrated below.

block of apartments built above a store with the first apartment at 16 feet and the second at 22 feet elevation.

Question 10.1

Make the assumption that the pattern of increase in height of each apartment continues at the same pattern. Fill out the rest of the chart.

Firstly, the question tells us to assume that the height increase from each apartment continues at the same pattern.

Another way to word this is, the change in height between each apartment is always the same.

So the change from apartment 2 to apartment 1 is 22 feet - 16 feet = 6 feet

So the change in height every time we go up one apartment will be 6 (feet). All we have to do everytime we go up one apartment is add 6 (feet). For this chart the units are included at the top, so we just need the number filled in.

1: 16

2: 22

3: 22 + 6=28

4: 28 + 6=34

5: 34 + 6=40

6: 40 + 6=46

7: 46 + 6=52

8: 52 + 6=58

Question 10.2

Trevor thinks he has created an equation that will tell you the height of the apartment, H, based off the number of the apartment, N.

Trevor's equation is: H = 10N + 6. Is this the correct equation?

Does this give the right height for apartment number 1 (or N = 1), what about for N = 2 and N = 3.

This may be one of the first times you have come across equations. It often helps to write the equation as a sentence.

Look at the symbols in the equation and find what those symbols mean. Write them in the equation

We have H, which symbolizes the height of the apartment and N which symbolizes the number of the apartment.

So substituting this in we get the height of the apartment = 10 x the number of the apartment + 6

Now, the question is asking us to check if the equation gives you the right height of apartment, H, when you put in the number of the apartment, N.

First we check for apartment number 1 or N = 1

H = (10 x 1) + 6 = 16

Checking what the question gives us we can see this is true. But it does not mean the equation is right, as it must work for all the apartment numbers (it must work for all values on N).

For N = 2 we have:

H = (10 x 2) + 6 = 26 which does not equal what the question gave us, 22.

And for N = 3 we have:

H = (10 x 3) + 6 = 36 which again, does not equal what we worked out through the pattern, 28.

So, to answer the question, no, Trevor's equation does not work.

Question 10.3

The correct equation is: H = 6N + 10. Verify this equation matches your table from above for N = 1, N = 2, and N = 3.

Now, the question is giving you the equation to calculate height, H, using the apartment number, N.

When it says “verify” it simply means to check that when you put the vaues for the apartment number, N that it gives you the corresponding height for that apartment, H.

For apartment number 1 or N=1, we sub in the value 1 into the equation where it says N.

H = 6N + 10

H = 6(1) + 10 = 6 + 10 = 16, which means H=16, [reminder: 10(1) is the same as writing 10 x 1] So, for N = 1, H = 16.

What the equation says is that, for apartment number 1, the height is 16 feet.

After we check the table from a), we can see this fits.

Now to check for N = 2, following the same process,

H = 6N + 10

H = 6(2) + 10 =

12 + 10 = 22

H = 22

So, for apartment number 2 or N=2, the height, H = 22

Checking our graph, we can see this is correct.

[A reminder, as you may notice the value of H changes every time. These algebraic equations give us answers to problems, and that answer is dependent on variables, which naturally change. There is not necessarily one value for an equation, like this example, the value of H does not stay the same. This reminder is just in case you were getting confused about why the value of H keeps changing!]

For the last one, N = 3.

H = 6N + 10

So, H = 6(3) + 10 = 28

H = 28

Checking our table we can see again this is correct. So we conclude that the given equation is correct.

Question 10.4

Using the formula in c), how high would apartment number 22 be? Show your work.

Start with what we know from the question,

Using the formula in c) we have H = 6N +10,

It is asking for the height of apartment number 22.

This fits perfectly with what the equation gives us,

The height of an apartment is 6 times the apartment number plus 10.

We know the apartment number is 22, or the way we have learned to write it and the way we have to write it: N=22

So, we want the height so simply follow the same process as in the previous question

H = 6N + 10

N = 22

H = 6(22) + 10 = 116 + 10 = 126

H = 126

So when N=22 (apartment number 22), H=126 (the height is 126 feet).

Question 10.5

We know the height of an apartment is 82 feet. Which apartment number must this be? Find the equation that gives us the apartment number, N, based on the variable height, H.

Start with what we know,

We know the apartment height is 82 feet, right this out with the algebraic symbols we have been using,

Height is 82 feet, or H = 82

We also know the equation from the previous question,

H = 6N + 10

Have a think about what we can do with these two pieces of information the question gives us?

What we can do is substitute what we know into the equation.

So put H = 82 into H = 6N + 10

(82) = 6N + 10

What we have now, is an equation with one variable symbol in it, N. An equation with only one variable is one that we can solve. We have to rearrange this equation to get our value for N.

If you mathematically do something to both sides of an equation it will still be true, a simple example is the equation,

1 = 1,

Say we multiply both sides by 3

3(1) = 3(1)

3 = 3

So, the equation still remains true, another example is

6 = 4 + 2

Say we divide both sides by 2

(6)/2 = (4+2)/2

6/2 = 4/2 +2/2

3 = 2 + 1 = 3

We can use these laws of mathematics to rearrange equations to suit our needs,

In this case we need to know the value of the apartment number, N,

We will need something that looks like N = (...)

So we need to use our basic mathematical tools of multiplication, division, addition and subtraction evenly on both sides of the equation to give us N = something,

So, again, putting H = 82 into H = 6N + 10 we get:

82 = 6N + 10,

We want a statement that says N = ... so we want to get all other numbers over to the other side of the equation, when doing this, it is generally easier to do the addition and subraction first.

Start with removing the 10 from the right side of the equation by subtracting 10, but remember, it has to be from both sides!

(82) - 10 = (6N + 10) - 10

82 - 10 = 6N + 10 - 10

[on the left 82 - 10 = 72 and on the right 6N + 10 - 10 = 6N]

72 = 6N

Now, we have to get the 6 over to the other side to leave us with just N, so we divide both sides by 6,

(72)/6 = (6N)/6

72/6 = (6/6)N

[on the left 72/6 is 12 and 6N/6 = 1N = N]

12 = N

And there we have our answer. N = 12,

So for H = 82 (when the height is 82 feet), N=12 (the apartment number is 12)

The second part of this question asks us to find the equation that gives us the apartment number, N, based on the variable height, H.

For this, again we need to rearrange our original equation to give us the apartment number, N based on the variable height, H. We use the same method we used in the first part of this method, only we dont have a numerical value for H, and it will remain as a symbol.

We want something that looks like N = ...

So, we have

H = 6N + 10

Just because there are now two variables (N & H) does not mean we can't use the same mathematical methods on both sides of the equation, to rearrange it for our needs.

Start again by subtracting both sides by 10

(H) - 10 = (6N + 10) - 10

H - 10 = 6N + 10 - 10

[ on the right, 6N + 10 - 10 = 6N]

H - 10 = 6N

Now as before, we divide both sides by 6 to give us an N on its own,

(H - 10) / 6 = N

So N = (H - 10)/6 or you can also write it as:

N
=
H - 10
6

N = (⅙)H - (10/6)

Note that 10/6 = 5/3 our equation is:

N = (⅙)H - (5/3)

A fun extra experiment to make sure this works, try putting in a value for H that we already know and see if it works,

Try H = 82, from earlier in question.

N = (H - 10)/6

N = (82 - 10)/6

N = 72/6

N = 12

Which, as we know, is correct!

Question 11

Lucy can do 60 push ups in 4 minutes. If she has been doing her push ups at a constant rate, how many push ups will she have done in 1 minute?

So, we know Lucy can do push ups at a rate of 60 push in 4 minutes.

This is the same as 60 push ups per 4 minutes, or 60 push ups / 4 minutes.

We want to know how many push ups she will have done in 1 minute.

Although the rate is per 4 minutes and not what you are probably used to i.e. per 1 minute, we still follow the same methods.

Take the rate and then multiply by the time,

60 push ups
4 minutes
x 1 minute

The minutes on top cancels the minutes on the bottom, we are left with

60 push ups / 4 which is:

15 push ups

Another way to think of it is, if we know she does 60 push ups in 4 minutes, and we want to know how many push ups she can do in 1 minute, we can divide 4 minutes by 4, to give us 1 minute. If we do this we must also divide the 60 push ups by 4. Which gives us 15 push ups.