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 $983042$