I can´t seem to do this

Question

At Irvś 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 tricycles does Irv have.

Answer 1

Lets clarify some terminology:

U-Unicycles =1 wheel

R-tandem bikes (regular bikes)= 2 wheels

T- tricycle = 3 wheels

He has a total of 115 wheels

We know that he has 32 Regular bikes (32 bikes> 2 wheels each> 64 wheels)

Number of bikes:

32+ U+T= 57

He has 57 helmets- one for each bike meaning he has 57 bikes total

32+U+T=57

U+T=25

U=25-T

Number of Wheels:

64+ U+ 3T= 115

U+3T=51

Substitute for U

25-T+3T=51

2T+25=51

2T=26

T=13 tricycles

13 tricycles * 3 wheels= 39 wheels

Substitute T back in here:

U=25-T

U=25-13

U=12 Unicycles

12 unicycles *1 wheel= 12 wheels

64 + 39+ 12 = 115 wheels

12+13+ 32= 57 Bikes

Hope this helps

Answer 2

The key to this problem is to assume that that the number of helmets equals the total number of bikes

of all types. Introduce the following variables:

u = the number of unicycles* ( Note the number of unicycles equals the number of tandem bikes )

t = the number of tandem bikes

r = the number of regular bikes = 32

tr = the number of tricycles

We have two equations one for helmets (total bikes) and one for the number of wheel

( Note unicycles have 1 wheel, tandem bikes 3 wheels, regular bikes 2 wheels & tricycles 3 wheels.)

u + t + r + tr = 57 and

u + 3t + 2r + 3tr = 115

Substituting t = u and r = 32 we have

2u + 32 + tr = 57 or tr = 25 - 2u Substituting this for tr in the second equation yields the following

4u + 64 + 75 - 6u = 115 or

-2u = - -24 or u = 12 So t =12 Using the first equation we see that tr = 1