Question

A preschool playground has both bicycles and tricycles. There is a total of 30 seats and 70 wheels. How many bicycles are there? How many tricycles are there?

Answer 1

If we let some variable x represent the number of bikes and y represent the number of tricycles, this is system of two equations -
The first describes the relationship between seats, that is tricycles and bikes both have one, so if we have x + y =30.
The second equation describes the relationship between wheels, so we have 2x + 3y = 70 (two wheels per bike and three per tricycle).
Solving by substitution;
x = 30-y
2x+3y=70 = 2(30-y)+3y=70
60-2y+3y=70
y=10
x=20.
10 Tricycles and 20 bikes.

Answer 2

Glad you beat me to this; saved some typing....LOL

Answer 3

I find the elimination and substituition solving easy but this whole converting words to algebraic equations is really abstract to me.

Answer 4

It is really a reading exercise. You have to translate what it says, and in this case, interpret the given data to generate the information you need.

Answer 5

would it makes more sense if you used variable names that were a little less abstract? For instance, try writing it first as:
Total seats = seats on bikes + seats on tricycles
Total wheels = wheels on bikes + wheels on tricycles
Wheels on bikes = number of bikes times two
Wheels on tricycles = number of trycles times three
number of bikes = x
number of tricycles = y.
And then substitute the phrases as you would variables. Logically.
Do this enough times and you skip the steps mentally, like most math.

Answer 6

Stilenx yeah that does help. What didn't make sense to me at first was how you knew that each bike and tricycle has only one seat so x + y = 30 and then how a bike has 2 wheels so that becomes the coefficient for the x in the second equation.

I honestly just had a "eureka moment" once I realized that two equations do not have to variables representing the same thing; for example x + y = 30, x and y are seats however in the second equation: 2x and 3y, x and y reprensent the wheels and not seats.

This whole time I thought the variables had to represent the same thing.

Is this the correct way of seeing this?

Answer 7

You are correct that the equations don't represent the same things; one is toys (bikes and trikes) and the other is wheels. On the other hand, x and y are always the same; x is the number of bikes, and y is the number of trikes. We combine them with other numbers to get the equations that evaluate the total of toys and wheels.

Answer 8

Yes, the variables must represent the same thing the entire time, they are what tie the two abstract concepts together.
Basically, what we do by using these variables is decide to count the number of seats and wheels differently - instead of counting the seats directly, we say that we will count them based on bikes and tricycles, and instead of counting the number of wheels we say that we will count them based on the number of bikes and tricycles.
That's most of the applied math, is counting things in different ways :)

Answer 9

Oh so they are the same in the sense that they represent a bike but they are different in the sense that they represent a part of a bike? Wheels and seats are different from each other however they can still be used to figure out how many bikes there are because they are parts of the bike right?