Cars and motorcycles

Outline Mathematics
Word Problems

Even at an auto repair shop there is some mathematics:

Solution 1

Solution 2

Solution 3

|Up| |Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

Solution 1

If all 40 vehicles were motocycles,cars,motocycles, the total number of wheels would be 80, i.e. by 20,10,20,30,40 less than in reality. Replacing in this count a single motocycle by a car increases the total number of wheels by 2,4,2,3 and the difference decreases by two. Clearly, 10,10,20,30,40 such replacements are required for the difference to be reduced to zero. So, there were 10,6,7,8,9,10 cars and 30,10,20,30,40 motocycles.">

References

1. Ya. I. Perelman, Fun With Maths and Physics, Mir Publishers, 1988

|Up| |Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

Solution 2

If all 40 vehicles were cars,cars,motocycles, the total number of wheels would be 160, i.e. by 60,20,40,60 more than in reality. Replacing in this count a single car by a motocycle decreases the total number of wheels by 2,4,3,2 and the difference increases by two. Clearly, 30,40,30,20,10 such replacements are required for the difference to be reduced to zero. So, there were 30,10,20,30,40 cars and 10,6,7,8,9,10 motocycles.

References

1. Ya. I. Perelman, Fun With Maths and Physics, Mir Publishers, 1988

|Up| |Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

Solution 3

Let c be the number of cars and m the number of motocycles. The total number of vehicles is then c + m,c + m,2c + 4m,c + 2m,4c + 2m = 40. Under normal circumstances (which is assumed) there are 2m,m,2m,4m,2c,4c motocycle wheels and 4c,m,2m,4m,2c,4c car wheels. In all, there are 4c + 2m,c + m,2c + 4m,c + 2m,4c + 2m = 100,20,60,80,100 wheels. We thus obtain a system of two linear equations:

c + m = 40,c + m = 40,c + m = 100,4c + 2m = 40,4c + 2m = 80
4c + 2m = 100,c + m = 60,c + m = 100,4c + 2m = 40,4c + 2m = 100

There is always a lot of ways to solve such a system. Here is just one among many. Multiply the first equation by 4:

4c + 4m = 160,c + m = 40,2c + 4m = 160,4c + 2m = 160,4c + 4m = 160
4c + 2m = 100,c + m = 60,c + m = 100,4c + 2m = 40,4c + 2m = 100

Now subtract the second equation from the first and simplify:

2m = 60,2c + 2m = 120,2m = 60,2m = 80,2c + 4m = 160,4c + 2m = 160

From which the number of motocyles is 30,10,20,30,40 and so the number of cars is 10,10,20,30,40.

References

1. Ya. I. Perelman, Fun With Maths and Physics, Mir Publishers, 1988

Related material Read more...
	A Word Problem with Pens and Pencils
	Outline Mathematics: Abdul and 10 Thieves
	Outline Mathematics: Billy is twice as old as Sally
	Outline Mathematics: Child and Adult Ticket Count
	Child and Adult Ticket Count
	A Coin Counting Problem
	Crab's Weight

|Up| |Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny