Math Help Forum - View Single Post

Soroban · #2 May 14th, 2008, 10:23 PM

Hello, fishnprincipal!

Quote:

At Irv's cycle rental shop, he rents all kinds of cycles:
unicycles, tandem bikes, regular bikes, and even tricycles for little kids.
He has 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?

$\begin{array}{c|cc} & \text{Helmets} & \text{Wheels} \\ \hline \text{Unicycles (U)} & 1 & 1 \\ \text{Tandems (T)} & 2 & 2 \\ \text{Bicycle (B)} & 1 & 2 \\ \text{Tricycles (R)} & 1 & 3 \\ \hline \text{Totals:} & 57 & 115 \end{array}$

We have: . $U + 2T + B + R \:=\:57$
. . . and: . $U + 2T + 2B + 3R \:=\:115$

We are told that: . $U = T,\:B = 32$

The equations becomes: . $\begin{array}{cccccccc} U + 2U + 32 + R &=&57 & \Rightarrow & 3U + R &=&25 & {\color{blue}[1]}\\ U + 2U + 64 + 3R &=&115 & \Rightarrow & 3U + 3R &=&51 & {\color{blue}[2]} \end{array}$

Subtract [1] from [2]: . $2R \:=\:26 \quad\Rightarrow\quad\boxed{ R \:=\:13}$

Substitute into [1]: . $3U + 13 \:=\:25 \quad\Rightarrow\quad\boxed{U \:=\:4} \quad\Rightarrow\quad\boxed{T \:=\:4}$

Therefore: .4 unicycles, 4 tandems, and 13 tricycles.