To solve for \( x \) and \( y \) such that the fractions \( \frac{614}{x7} = \frac{614}{24y} \) are equivalent, we first need to express these fractions in a more manageable form.
-
The first fraction is \( \frac{614}{x7} \):
- Here, \( x7 \) represents the number formed by the digit \( x \) followed by 7, which can be expressed as \( 10x + 7 \).
-
The second fraction is \( \frac{614}{24y} \):
- Here, \( 24y \) represents the number formed by 24 followed by \( y \), which can be expressed as \( 240 + y \).
Now we can set the fractions equal to each other:
\[ \frac{614}{10x + 7} = \frac{614}{240 + y} \]
Since the fractions are equal, we can cross-multiply:
\[ 614(240 + y) = 614(10x + 7) \]
Since both sides have \( 614 \), we can simplify this to:
\[ 240 + y = 10x + 7 \]
Next, let's rearrange the equation to find \( y \):
\[ y = 10x + 7 - 240 \] \[ y = 10x - 233 \]
Now, for the three fractions to be equivalent and since both \( x \) and \( y \) must be digits (ranging from 0 to 9), let's explore possible values for \( x \).
-
If \( x = 0 \): \[ y = 10(0) - 233 = -233 \quad \text{(Not a valid digit)} \]
-
If \( x = 1 \): \[ y = 10(1) - 233 = -223 \quad \text{(Not a valid digit)} \]
-
If \( x = 2 \): \[ y = 10(2) - 233 = -213 \quad \text{(Not a valid digit)} \]
-
If \( x = 3 \): \[ y = 10(3) - 233 = -203 \quad \text{(Not a valid digit)} \]
-
If \( x = 4 \): \[ y = 10(4) - 233 = -193 \quad \text{(Not a valid digit)} \]
-
If \( x = 5 \): \[ y = 10(5) - 233 = -183 \quad \text{(Not a valid digit)} \]
-
If \( x = 6 \): \[ y = 10(6) - 233 = -173 \quad \text{(Not a valid digit)} \]
-
If \( x = 7 \): \[ y = 10(7) - 233 = -163 \quad \text{(Not a valid digit)} \]
-
If \( x = 8 \): \[ y = 10(8) - 233 = -153 \quad \text{(Not a valid digit)} \]
-
If \( x = 9 \): \[ y = 10(9) - 233 = -143 \quad \text{(Not a valid digit)} \]
It seems that there are no valid numbers for \( x \) that yield a digit for \( y \) under the constraints provided.
To find valid values of \( x \) and \( y \), we would need to set \( y \) to be a digit within 0-9, and then adjust \( x \) accordingly.
If we try \( y \) from 0 to 9, we can also find corresponding \( x \):
For example, if \( y = 0 \): \[ 0 = 10x - 233 \Rightarrow 10x = 233 \Rightarrow x = 23.3 \quad \text{(Not valid)} \]
Continuing this for other values of \( y \):
We repeat the calculation until we find a valid pair.
In conclusion, you will find numbers that satisfy conditions or notice a discrepancy leading to the realization that no valid (x, y) can be found.
1 answer