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My dughter's homework is very difficult. Can you please help? It goes like this: "At Irv's Cycle Rental Shop, Irv rents all kinds of cycles: unicycles, tandem bikes, regular bikes, and even tricycles for little kids. He parks all the cycles in front of his shop with a helmet for each rider strapped to the cycles. This morning Irv counted 57 helmets and 115 wheels parked in front of his store. He knows he has an equal number of unicycles and tandem bikes. He also knows that he has 32 regular bikes. How many unicycles, tandem bikes, and tricylcles does Irv have?" Please help her. Tell me how you got the answer. And not just an answer. If you don't know the answer but know how to work it. Please tell me that too. Thanks.

2007-09-27 11:41:20 · 2 answers · asked by tagamargos 1 in Science & Mathematics Mathematics

2 answers

Define variables for each of the cycle types:
Let N be the number of unicycles [1 helmet (rider) and 1 wheel each]
Let N also be the number of tandem bikes [2 helmets (riders) and 2 wheels each] -- the reason we can use the same variable is that we are told that the number of each is the same.
Let T be the number of tricycles [1 helmet (rider) and 3 wheels each]
You don't need a variable for bicyles because you are told you have 32 [32 helmets and 64 wheels]

Now you can form two equations, one for number of helmets, and another for the number of wheels.

Number of helmets = 57
This includes 32 for bicycles, N for unicycles, 2N for tandem bikes and T for tricycles:
N + 2N + T + 32 = 57

Simplifying by grouping the N terms and subtracting 32 from both sides:
3N + T = 25

Next form an equation for the number of wheels (115):
N for the unicycles
2N for the tandem bikes
64 for the bicycles
3T for the tricycles

All together:
N + 2N + 64 + 3T = 115

Simplifying by grouping the N terms and subtracting 64 from both sides:
3N + 3T = 51

So the two equations are:
3N + 3T = 51
3N + T = 25

Notice how you have a common 3N. If you subtract the bottom equation from the top the 3N will cancel out:
2T = 26

Divide both sides by 2:
T = 13

So we have the number of tricycles. Plug this back into one of the equations:
3N + T = 25
3N + 13 = 25
3N = 12
N = 4

So you have:
4 unicycles = 4 helmets / 4 wheels
4 tandem bikes = 8 helmets / 8 wheels
32 bicycles = 32 helmets / 64 wheels
13 tricycles = 13 helmets / 39 wheels

You should always double-check, so let's add up helmets and wheels to confirm we get 57 and 115 respectively:
4 + 8 + 32 + 13 = 57 helmets
4 + 8 + 64 + 39 = 115 wheels

2007-09-27 12:01:08 · answer #1 · answered by Puzzling 7 · 2 0

U = unicycles (1wheel, 1 helmet)
T = Tandem bikes (2 wheels, 2 helmets)
B = Regular bikes (2 Wheels, 1 helmet)
i = Tricycles (3 wheels, 1 helmet)
H = # of helmets = 57
W = # of wheels = 115

Numbers of :
U = T
B = 32 (there are 64 wheels form Bikes, and 32helmets)

We can start by expressing both the 57 and the 115 in equations (terms of T. B, U, and i), using x, y, and Z as the unknown quantities of T(same as U), B, and i

Helmets Equation:
57 = xT(*2) +xU, +yB +Zi
... T is (*2) [means multiplied by two] because there are 2 helmets for every Tandem bike

Wheels Equation:
115 = xT(*2) +xU + yB(*2) +Zi(*3)
... T is (*2), B is (*2) and i is (*3) cause that is how man wheels there are per bike of that type

Once you get the Wheels and Helmets equations the mathematics should be (fun and) workable by your daughter.

2007-09-27 18:56:15 · answer #2 · answered by David F 5 · 0 2

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