Calculate $98×93$

__Solution__

Select the closest base (a power of $10$). In this case we can select $100$ as a base. Subtract base from these numbers. That is $98-100=-2$ and $93-100=-7.$ So write it as

$98\qquad -2\\93\qquad -7$

Left side of the answer will be the diagonal sum including their signs. That is $98-7=91.$ So $91$ will be the left side of the product. (You can take the other diagonal sum also, that is $93-2=91.$ Always these diagonal sums will be same).

To find the right side, just multiply the differences including their signs. That is $-2×-7=14$

So we got that left side of the product is $91$ and right side is $14.$ So answer is $9114$

Calculate $96×112$

__Solution__

Use the same method. Let's select $100$ as the base (close to both the numbers). Subtract base from these numbers. That is $96-100=-4$ and $112-100=12.$ So write it as

$96\qquad -4\\112\qquad 12$

Take the diagonal sum to get the left side of the product. That is $96+12=108$ (Or $112-4=108$).

For the right side, find the product of the differences. That is $-4×12=-48.$ Since it is -ve, we need to make it as +ve. For this borrow $1$ from our left side $108.$ This borrowed $1$ becomes $100$ (because base is $100$) and our right side becomes $100+(-48)=52.$

Since we borrowed one from left side, left side becomes $107$

So answer is $10752$

Calculate $103×115$

__Solution__

Use the same method. Let's select $100$ as the base (close to both the numbers). Subtract base from these numbers. That is $103-100=3$ and $115-100=15.$ So write it as

$103\qquad 3\\115\qquad 15$

Take the diagonal sum to get the left side of the product. That is $103+15=118$ (Or $115+3=118$).

For the right side, find the product of the differences. That is $3×15=45$

So answer is $11845$

Calculate $122×89$

__Solution__

Use the same method. Let's select $100$ as the base (close to both the numbers). Subtract base from these numbers. That is $122-100=22$ and $89-100=-11.$ So write it as

$122\qquad 22\\89\qquad -11$

Take the diagonal sum to get the left side of the product. That is $122+(-11)=111$ (Or $89+22=111$).

For the right side, find the product of the differences. That is $22×(-11)=-242.$ Since it is -ve, we need to make it as +ve. For this borrow $3$ from our left side $111.$ This borrowed $3$ becomes $300$ (because base is $100$ and $3×100=300$) and our right side becomes $300+(-242)=58$

Since we borrowed $3$ from left side, left side becomes $108$

So answer is $10858$

Calculate $1024×989$

Use the same method. Let's select $1000$ as the base (close to both the numbers). Subtract base from these numbers. That is $1024-1000=24$ and $989-1000=-11.$ So write it as

$1024\qquad 24\\989\qquad -11$

Take the diagonal sum to get the left side of the product. That is $1024+(-11)=1013$ (Or $989+24=1013$).

For the right side, find the product of the differences. That is $24×(-11)=-264.$ Since it is -ve, we need to make it as +ve. For this borrow $1$ from our left side $1013.$ This borrowed $1$ becomes $1000$ (because base is $1000$) and our right side becomes $1000+(-264)=736$

Since we borrowed $1$ from left side, left side becomes $1012$

So our answer is $1012736$

Calculate $997×986$

Use the same method. Let's select $1000$ as the base (close to both the numbers). Subtract base from these numbers. That is $997-1000=-3$ and $986-1000=-14.$ So write it as

$997\qquad-3\\986\qquad -14$

Take the diagonal sum to get the left side of the product. That is $997+(-14)=983$ (Or $986-3=983$).

For the right side, find the product of the differences. That is $(-3)×(-14)=42.$ But here $42$ has only $2$ digits whereas our base $1000$ has three zeros. So write $42$ as $042$

So our answer is $983042$

Later, 1 was borrowed from 108 for making RHS positive.

hence LHS became 107.

This can be done using base 100 or 1000. But it will hardly help in solving speedily.

Using base 100

405 :305

397 :297

Left side answer = 702 (405+297 or 397+305)

Right side answer = 90585

Final answer

Right side answer = 85 ( 2 left side of the number 90585)

Left side answer = 1607 (702+ 905)

**1607 85**

A better method is to apply the formula (x + a)(x –b) = x

^{2}+ x(a-b) – ab

405= (400+5)

397= (400-3)

405 × 397 = (400+5)×(400-3)=1600+400(5-3)-15=160000+800-15=160785

In this sum,why we have borrowed 1 rather than 3?

If we have borrowed 3 then it may become 300-264=36.

36 is also +ve number know.Then why we have borrowed 1 ?

So, borrowed 1 becomes 1000

No need to borrow 3 as it becomes 3000

557 : -443 (1000-443)

613 : -387(1000-387)

Left side of the answer will be the diagonal sum including their signs : 170(557-387 or 613-443)

Right side of the answer will be the product of the differences : -443 × -387 = 171441 (First three digits will add up with the left side of the answer)

So the final answer is 341(170+171) 441 (Left side answer Right side answer) ie 341 441

Although this procedure is applicable but it is rather lengthy for this example.

I feel it is more appropriate for nos. which are nearer to 10,100,1000, etc.

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