Evaluate (-25) x (-12) + (-25) x (-8) using distributive property

Question

Evaluate $(-25) \times (-12) + (-25) \times (-8)$ using the distributive property of multiplication over addition.

Solution — Step by Step

Look at both terms:

First term: $(-25) \times (-12)$
Second term: $(-25) \times (-8)$

Both terms have $(-25)$ as a common factor. This is exactly the setup for the distributive property: $a \times b + a \times c = a \times (b + c)$ .

(-25) \times (-12) + (-25) \times (-8) = (-25) \times \big[(-12) + (-8)\big]

We factored out $(-25)$ from both terms. Now we add the numbers inside the bracket.

(-12) + (-8) = -20

(Adding two negative numbers: add the absolute values, keep the negative sign)

(-25) \times (-20) = 500

Negative × Negative = Positive

The product of two negative integers is always positive.

Answer: 500

$(-25) \times (-12) = 300$

$(-25) \times (-8) = 200$

$300 + 200 = 500$ ✓

Both methods give the same answer, confirming the result.

Why This Works

The distributive property says $a(b + c) = ab + ac$ . We used it in reverse here — going from two separate products to one product. This is called factoring or using the distributive property “backwards.”

The reason this saves calculation effort: instead of doing two multiplications and one addition, we do one addition (simpler: $-12 + (-8) = -20$ ) and one multiplication ( $-25 \times -20 = 500$ ).

In general, when you spot a common factor in an expression, extracting it first almost always simplifies the computation.

Alternative Method

Direct calculation (without the distributive property):

$(-25) \times (-12) = 300$ (since $25 \times 12 = 300$ and negative × negative = positive)
$(-25) \times (-8) = 200$ (since $25 \times 8 = 200$ )
$300 + 200 = 500$

The distributive method is faster and more elegant, especially when numbers are larger.

For CBSE Class 7, whenever you see a pattern like $a \times b + a \times c$ or $(-a) \times (-b) + (-a) \times (-c)$ , immediately think “distributive property.” It is tested explicitly in board exams with the instruction “use distributive property” — showing the intermediate step of factoring out the common factor is mandatory for full marks.

Common Mistake

Students often forget the sign rules for multiplying integers. $(-25) \times (-12) = +300$ , NOT $-300$ . Two negatives multiply to give a positive. Also, some students mistakenly compute $(-25) \times [(-12) + (-8)] = (-25) \times (-20)$ correctly but then write the final answer as $-500$ . Remember: negative times negative is positive. Final answer is $+500$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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