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.
Let the total number of skateboards be $4x$ and the total number of scooters be $3y$. Then, $\frac{1}{4}(4x) = \frac{2}{3}(3y)$, so $x=3y$. Then, the number of red skateboards is $x=\frac{3y}{3} = y$, and the number of red scooters is $y$. The total number of skateboards and scooters is $4x + 3y = 4(3y) + 3y = 15y$. The total number of red skateboards and scooters is $2y + y = 3y$. Therefore, the fraction of the total that is red is $\boxed{\frac{3y}{15y}} = \boxed{\frac{1}{5}}$.
To find the fraction of the total skateboards and scooters that are red, we need to determine the common denominator between 1/4 and 2/3.
The least common multiple (LCM) of the denominators 4 and 3 is 12.
To convert 1/4 to a fraction with a denominator of 12, we multiply the numerator and denominator by 3: (1/4) * (3/3) = 3/12.
To convert 2/3 to a fraction with a denominator of 12, we multiply the numerator and denominator by 4: (2/3) * (4/4) = 8/12.
Since we know there are equal numbers of red skateboards and red scooters, we can add their fractions together: 3/12 + 8/12 = 11/12.
Therefore, 11/12 of the total skateboards and scooters (together) are red.
To find the fraction of the total skateboards and scooters that are red, we first need to determine the individual fractions of red skateboards and red scooters.
Let's assume the total number of skateboards in the shop is represented by the variable 's', and the total number of scooters is represented by the variable 'c'.
Given that 1/4 of the skateboards are red, we can express the fraction of red skateboards as 1/4 of 's', which is (1/4) * s.
Similarly, since 2/3 of the scooters are red, we can express the fraction of red scooters as 2/3 of 'c', which is (2/3) * c.
According to the problem, the number of red skateboards is equal to the number of red scooters, so we can set up an equation:
(1/4) * s = (2/3) * c
To simplify this equation and remove the fractions, we can multiply both sides by the least common denominator of 4 and 3, which is 12:
12 * (1/4) * s = 12 * (2/3) * c
3s = 8c
Now, we can express the fraction of red skateboards and scooters as the sum of the fractions of red skateboards and red scooters divided by the total number of skateboards and scooters:
(red skateboards + red scooters) / (total skateboards + total scooters)
= [(1/4) * s + (2/3) * c] / (s + c)
Substituting the equation we found earlier, we have:
[ (1/4) * s + (2/3) * c ] / (s + c)
= [ (1/4) * s + (2/3) * c ] / (3s/3 + c)
= [ (1/4) * s + (2/3) * c ] / (3s + 3c) / 3
= [ 3 * (1/4) * s + 3 * (2/3) * c ] / (3s + 3c) / 3
= [ 3s/4 + 2c ] / (3s + 3c) / 3
= (9s/4 + 6c) / (3s + 3c) / 3
= (9s/4 + 6c) / 3s + 3c
Therefore, the fraction of the total skateboards and scooters together that are red is (9s/4 + 6c) / (3s + 3c).