Multiply Near a Base (200 to 900)

Multiply two numbers close to a base of 200 to 900 using the Indian mental-math method, with each step of the cross-sum and correction shown.

Input

A way to multiply two numbers near a base of 200 to 900 in your head, using how far each one sits from the base.

Base

First number

Second number

Choose a base from 200 to 900. The closer the two numbers are to the base, the easier the mental math.

Answer for 308 times 304

93,632

Cross-sum

312

Main part (times base)

93,600

Correction (product of deviations)

Step by step

Find the deviations

Against the base 300, 308 has a deviation of +8 and 304 has +4.

Take the cross-sum

Add +4 to 308 to get 312. Adding +8 to 304 gives the same 312.

Multiply by the base

Multiply the cross-sum 312 by the base 300 to get 93,600.

Multiply the deviations

Multiply the deviation 8 by 4 to get the correction 32.

Add for the answer

Add the correction 32 to the main part 93,600 to get 93,632.

Calculation breakdown

Item	Value
Base	300
Deviation of 308	+8
Deviation of 304	+4
Cross-sum	312
Main part (times base)	93,600
Correction (product of deviations)	32
Answer (product)	93,632

Check: 308 times 304 = 93,632, the same as ordinary multiplication.

The formula used is a1 times a2 = b times (a1 plus d2) plus d1 times d2, where b is the base and d1, d2 are the deviations.

Pick a base b (a multiple of 100 from 200 to 900) and enter two numbers a1 and a2 near it.
Find each deviation from the base: d1 = a1 minus b and d2 = a2 minus b.
Take the cross-sum (a1 plus d2, which equals a2 plus d1) and multiply it by the base b for the main part.
Add the product of the deviations d1 times d2 as the correction to get a1 times a2.
As a formula, a1 times a2 = b times (a1 plus d2) plus d1 times d2. The trick is most useful when both numbers are close to the base.

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