Problem Solving with Rates & Patterns


Algebra is a part of mathematics in which letters and other general symbols are used as a language to represent numbers and quantities in formulae 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.

The solutions in the above animation are repeated with a more detailed explanation below.

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?

In 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

The solutions in the above animation are repeated with a more detailed explanation below.

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 an 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 2.2

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 above

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 into the fundamentals of Algebra as we start to increase the complexity of the relationship between sentences and words and symbolist that in equations.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 3

A motorbike and a car are driving towards each other down a single track road that is 150 miles long. They 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.

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?

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 150 mile long road?”

If they have each started at each end of the road, 150 miles apart and meet at some point, 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 to imagine the car and motorbike have met, 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, 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.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 4.1

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 4.2

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.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 5.1

Sam has 28 treats that he wants to divide evenly for 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 5.2

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

The solutions in the above animation are repeated with a more detailed explanation below.

Question 6

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

The solutions in the above animation are repeated with a more detailed explanation below.

Question 7.1

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 7.2

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.


Question 7.3

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.