If $9 @ 3 = 108$ and $10 @ 2 = 80$, what is $7 @ 1$?
Operator Rule Identification
We are given a mathematical operator '@' with two examples:
- $9 \text{ @ } 3 = 108$
- $10 \text{ @ } 2 = 80$
We need to determine the rule governing this operator.
Deriving the Operator Logic
Let's analyze the first equation, $9 \text{ @ } 3 = 108$. We can observe that the product of the two numbers is $9 \times 3 = 27$. To get 108, we multiply 27 by 4 ($27 \times 4 = 108$).
Let's test this potential rule, $a \text{ @ } b = (a \times b) \times 4$, with the second equation, $10 \text{ @ } 2 = 80$.
Here, $a=10$ and $b=2$. Applying the rule:
$(10 \times 2) \times 4 = 20 \times 4 = 80$.
The rule $a \text{ @ } b = 4 \times a \times b$ holds true for both given examples.
Calculating the Final Value
Now, we apply the confirmed rule $a \text{ @ } b = 4 \times a \times b$ to find the value of $7 \text{ @ } 1$.
Here, $a=7$ and $b=1$.
Calculation:
$7 \text{ @ } 1 = (7 \times 1) \times 4$
$7 \text{ @ } 1 = 7 \times 4$
$7 \text{ @ } 1 = 28$
Result
The value of $7 \text{ @ } 1$ is 28.
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