Lesson Problem on two-wheel and three-wheel bicycles

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Problem on two-wheel and three-wheel bicycles

Problem

There were two-wheel and three-wheel bicycles on sale.
In all, 7 bicycles were sold that have 15 wheels altogether.
How many two-wheel and how many three-wheel bicycles were sold?

You can solve this problem in three different ways.
If you are familiar with systems of linear equations, you can reduce the problem to the system of two linear equations in two unknowns and solve it.
This way is implemented in the Solution 1 below.
You can also reduce the problem to one equation with one unknown and solve it. This is done in the Solution 2 below.
Alternatively, you can solve the problem simply applying logical reasoning and not using equations at all. This is done in the Solution 3 below.

Solution 1

Let x be the number of two-wheel bicycles and y be the number of three-wheel bicycles sold in the sale.
If you count the bicycles you get the equation
x + y = 7.
If you count the wheels you get the equation
2x + 3y = 15.

So, you have the system of two equations with two unknowns
system+%28x+%2B+y+=+7%2C%0D%0A2x+%2B+3y+=+15%29%0D%0A

system+%28x+%2B+y+=+7%2C%0D%0A2x+%2B+3y+=+15%29%0D%0A

To solve this system of equations multiply the first equation by 2 and subtract the obtained equation from the second one.
You will get, step by step,
system+%282x+%2B+2y+=+14%2C%0D%0A2x+%2B+3y+=+15%29%0D%0A

system+%282x+%2B+2y+=+14%2C%0D%0A2x+%2B+3y+=+15%29%0D%0A

3y - 2y = 15 - 14,
y = 1.

So, one three-wheel bicycle was sold.
Hence, the number of two-wheel bicycles was 7-1=6.

Let us check the total number of the wheels. You have altogether
6*2 +1*3 = 12 + 3 = 15 wheels.

Answer. There were 6 two-wheel bicycles and 1 three-wheel bicycle sold on sale.

Solution 2

Let x be the number of two-wheel bicycles sold on sale.
Then the number of three-wheel bicycles sold on sale is 7 - x in accordance with the condition.
If you count the wheels you get the equation
2x + 3*(7-x) = 15.

To solve this equation open the brackets and combine like terms, step by step:
2x + 2*7 - 3x = 15,
-x + 21 = 15,
-x = 15 - 21,
-x = -6,
x = 6.

So, there were 6 two-wheel bicycles sold on sale.
Hence, the number of the three-wheel bicycles sold on sale was 7 - 6 = 1.

Let us check the total number of wheels. You have altogether
6*2 + 1*3 = 12 + 3 = 15 wheels.

You get the same answer as in the Solution 1.

Answer. There were 6 two-wheel bicycles and 1 three-wheel bicycle sold on sale.

Solution 3

Let us suppose for a moment that all bicycles sold on sale have two wheels each.
Under this assumption, the total number of wheels is 7*2 = 14.
This number is one less than 15 wheels given by condition.

It is clear that this wheel belongs to the three-wheel bicycle (in addition to that two wheels we just counted under the assumption).
This means that the number of three-wheel bicycles is 1.
Hence, the number of two-wheel bicycles is 7 - 1 = 6.

You get the same answer as in the solutions 1 and 2 above.

Answer. There were 6 two-wheel bicycles and 1 three-wheel bicycle sold in the sale.

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under the current topic Miscellaneous Word Problems of the section Word problems in this site.

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