88 more bicycles than unicycles, 5 times as many tricycles as unicycles, and 40 more bicycles than tricycles, how many are unicycles

1 answer

Let's define the number of unicycles as \( u \).

According to the problem, we can express the number of bicycles and tricycles in terms of \( u \):

  • The number of bicycles \( b \) is given by: \[ b = u + 88 \]

  • The number of tricycles \( t \) is given by: \[ t = 5u \]

  • We also know that the number of bicycles is 40 more than the number of tricycles: \[ b = t + 40 \]

Now we can substitute the expressions for \( b \) and \( t \) into the equation \( b = t + 40 \):

Substituting \( t = 5u \) into the equation gives: \[ b = 5u + 40 \]

Now we have two expressions for \( b \):

  1. \( b = u + 88 \)
  2. \( b = 5u + 40 \)

We can set these equal to each other: \[ u + 88 = 5u + 40 \]

Now, we can solve for \( u \): \[ 88 - 40 = 5u - u \] \[ 48 = 4u \] \[ u = 12 \]

So, the number of unicycles is \( \boxed{12} \).

Verification

Let's verify the solution:

  • If \( u = 12 \), then:
    • The number of bicycles \( b \) is: \[ b = 12 + 88 = 100 \]

    • The number of tricycles \( t \) is: \[ t = 5 \times 12 = 60 \]

Now we check if the relationship holds:

  • According to the problem:
    • Bicycles = Tricycles + 40
    • \( 100 = 60 + 40 \) is true.

Thus, the solution is verified, and the number of unicycles is indeed \( \boxed{12} \).