In the FreeWheelin' Skate Shop, exactly 1/4 of the skateboards are red and exactly 2/3 of the scooters are red. If there are equal numbers of red skateboards and red scooters in the shop, what fraction of the total skateboards and scooters together are red? Express your answer as a simplified fraction.
anonymous
Let's say there are $x$ skateboards in the shop. Thus, there are $x\cdot\frac{1}{4}=\frac{1}{4}x$ red skateboards and $\frac{2}{3}x $ red scooters. Since the number of red skateboards is equal to the number of red scooters, this means that, \begin{align*}
\frac{1}{4}x&=\frac{2}{3}x
\\\Rightarrow\qquad12\cdot\frac{1}{4}x&=12\cdot\frac{2}{3}x
\\\Rightarrow\qquad3x&=8x
\\\Rightarrow\qquad-5x&=0
\end{align*}Thus, $x=0.$ A skateboard and a scooter are both defined as non-zero, so both $x$ and $0$ are defined as zero. This means that there are zero skateboards, and hence, zero scooters, in the skate shop. Thus, the fraction of the total skateboards and scooters that are red is $\frac{0}{0}=\boxed{0.}$
To find the fraction of the total skateboards and scooters together that are red, we need to determine the ratio of red items to the total number of items.
Let's denote the total number of skateboards as S and the total number of scooters as C.
According to the information given, 1/4 of the skateboards are red. So, the number of red skateboards can be expressed as 1/4 * S.
Similarly, 2/3 of the scooters are red. Thus, the number of red scooters is 2/3 * C.
Since the number of red skateboards and red scooters are equal, we can set up an equation:
1/4 * S = 2/3 * C
To simplify this equation, let's multiply both sides by 12 (the least common multiple of 4 and 3) to eliminate the fractions:
12 * (1/4 * S) = 12 * (2/3 * C)
3S = 8C
Now, we can rewrite the equation as S = (8/3) * C.
So, the fraction of the total skateboards and scooters together that are red can be calculated by finding the total number of red items (red skateboards + red scooters) and dividing by the total number of items (skateboards + scooters):
Red fraction = (1/4 * S + 2/3 * C) / (S + C)
Since we know that S = (8/3) * C, we can substitute it into the formula:
Red fraction = (1/4 * (8/3) * C + 2/3 * C) / ((8/3) * C + C)
Simplifying further:
Red fraction = (2/3 + 2/3) / ((8/3) + 1)
Red fraction = (4/3) / (11/3)
Finally, we can simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator:
Red fraction = (4/3) * (3/11) = 4/11
Therefore, the fraction of the total skateboards and scooters together that are red is 4/11.
Let's assume that there are a total of x skateboards in the FreeWheelin' Skate Shop.
Since exactly 1/4 of the skateboards are red, the number of red skateboards would be (1/4) * x.
Similarly, if there are equal numbers of red skateboards and red scooters, then the number of red scooters would also be (1/4) * x.
Now, let's consider the scooters. We know that exactly 2/3 of the scooters are red. Let's assume there are y scooters in total. Therefore, the number of red scooters would be (2/3) * y.
Since the number of red scooters and red skateboards are equal, we can equate the two expressions:
(1/4) * x = (2/3) * y
To find the fraction of the total skateboards and scooters that are red, we need to determine the total number of skateboards and scooters:
Total = x (skateboards) + y (scooters)
Now, let's substitute the values of y from the equation above into the Total expression:
Total = x + (3/2) * (1/4) * x
Simplifying the expression:
Total = x + (3/8) * x
Total = (11/8) * x
To find the fraction of the total that is red, we need to divide the number of red skateboards and scooters by the Total:
Fraction = ((1/4) * x + (2/3) * y) / Total
Fraction = ((1/4) * x + (2/3) * (3/2) * (1/4) * x) / ((11/8) * x)
Simplifying the expression further:
Fraction = (1/4 + 1/4) / (11/8)
Fraction = (2/4) / (11/8)
Fraction = (2/4) * (8/11)
Fraction = 16/44
Fraction = 4/11
Therefore, the fraction of the total skateboards and scooters that are red is 4/11.