Apr 15, 2018

337

#### Explanation:

You have been given the four numbers 12, 7, 4 and 1 and the only other thing you can do with this is add, subtract, multiply or divide.

I used 12, 7 and 4 to multiply them together:

$12 \cdot 7 \cdot 4 = 336$

You could multiply by 1 as well, but this will just give you 336.
You do not want to subtract, otherwise, this will reduce the value, and division will give you 336, so the addition is all that's left.

$336 + 1 = 337$

So to show this, use brackets to separate the functions:
$\left(12 \cdot 7 \cdot 4\right) + 1 = 337$

Further explanation:

Multiplication is thought as repeated addition (copies of the same value) wheras division is used to separate the value into equal divisions. Subtraction reduces the number and addition increases. So the best method to get the largest possible value is by multiplication. If powers could be used, then this would make the end value even larger.

Apr 15, 2018

$\left(4 + 1\right) \cdot 12 \cdot 7 = 420$

#### Explanation:

The numbers $p$ and $q$ get the highest result by multiplication, if they are $\left(p , q\right) \ge 2$. Therefore, for $12 , 4 , 7$ the highest result is
$12 \cdot 4 \cdot 7 = 336$
But what do we do with the one? If we multiply, the result will be the same, just like when we divide. Subtraction will make the result samller, therefore we add $1$. Lets go through the possibilities:

$336 + 1 = 337$
Adding to the $12$:
$\left(12 + 1\right) \cdot 4 \cdot 7 = 364$
Adding to the $4$:
$12 \cdot \left(4 + 1\right) \cdot 7 = 420$
Adding to the $7$:
$12 \cdot 4 \cdot \left(7 + 1\right) = 384$
The smaller the product is we are adding $1$ to, the higher the result is going to be. Here is why:

Let $1 < a < b < c$:

$a \cdot b \cdot \left(c + 1\right) = a b c + a b$
$a \cdot \left(b + 1\right) \cdot c = a b c + a c$
$\left(a + 1\right) \cdot b \cdot c = a b c + b c$
$a b < a c < b c$