Use this to find (i) , (ii) , (iii) , (iv) , (v) and (vi) .

We will use the distributive property to simplify these multiplication problems. This property helps us break down complex multiplications into simpler ones. Step 1 — Multiplying by 101 (Part i) We want to calculate . We can write as . The distributive property is . Let , , and . So, we multiply by and then by . Then we add the results together. > Step 2 — Multiplying by 101 (Part ii) We need to find the product of and . Again, we write as . We use the distributive property. We multiply by and then by . Then we add these products. > Step 3 — Multiplying by 1001 (Part iii) We want to calculate . We can write as . We apply the distributive property. We multiply by and then by . Then we add the results. > Step 4 — Multiplying by 1001 (Part iv) Let us find the product of and . We express as . Using the distributive property, we multiply by and by . Then we add these two products. > Step 5 — Multiplying by 99 (Part v) We need to calculate . We can write as . The distributive property also works for subtraction: . Let , , and . So, we multiply by and then by . Then we subtract the second result from the first. > Step 6 — Multiplying by 999 (Part vi) Let us find the product of and . We write as . We use the distributive property for subtraction. We multiply by and by . Then we subtract the second product from the first. > Answer > (i) 8989 > (ii) 95849 > (iii) 266096831 > (iv) 1112111 > (v) 963666 > (vi) 23454522

Distributivity and Algebra | IT

Question 17

Use this to find (i) $89 \times 101$ , (ii) $949 \times 101$ , (iii) $265831 \times 1001$ , (iv) $1111 \times 1001$ , (v) $9734 \times 99$ and (vi) $23478 \times 999$ .

Solution

We will use the distributive property to simplify these multiplication problems. This property helps us break down complex multiplications into simpler ones.

Step 1 — Multiplying by 101 (Part i)

We want to calculate $89 \times 101$ . We can write $101$ as $100 + 1$ . The distributive property is $a \times (b+c) = a \times b + a \times c$ . Let $a = 89$ , $b = 100$ , and $c = 1$ . So, we multiply $89$ by $100$ and then by $1$ . Then we add the results together.

$89 \times 101 = 89 \times (100 + 1)$

$= (89 \times 100) + (89 \times 1)$

$= 8900 + 89$

$\boxed{8989}$

Diagram 1

Step 2 — Multiplying by 101 (Part ii)

We need to find the product of $949$ and $101$ . Again, we write $101$ as $100 + 1$ . We use the distributive property. We multiply $949$ by $100$ and then by $1$ . Then we add these products.

$949 \times 101 = 949 \times (100 + 1)$

$= (949 \times 100) + (949 \times 1)$

$= 94900 + 949$

$\boxed{95849}$

Step 3 — Multiplying by 1001 (Part iii)

We want to calculate $265831 \times 1001$ . We can write $1001$ as $1000 + 1$ . We apply the distributive property. We multiply $265831$ by $1000$ and then by $1$ . Then we add the results.

$265831 \times 1001 = 265831 \times (1000 + 1)$

$= (265831 \times 1000) + (265831 \times 1)$

$= 265831000 + 265831$

$\boxed{266096831}$

Diagram 2

Step 4 — Multiplying by 1001 (Part iv)

Let us find the product of $1111$ and $1001$ . We express $1001$ as $1000 + 1$ . Using the distributive property, we multiply $1111$ by $1000$ and by $1$ . Then we add these two products.

$1111 \times 1001 = 1111 \times (1000 + 1)$

$= (1111 \times 1000) + (1111 \times 1)$

$= 1111000 + 1111$

$\boxed{1112111}$

Step 5 — Multiplying by 99 (Part v)

We need to calculate $9734 \times 99$ . We can write $99$ as $100 - 1$ . The distributive property also works for subtraction: $a \times (b-c) = a \times b - a \times c$ . Let $a = 9734$ , $b = 100$ , and $c = 1$ . So, we multiply $9734$ by $100$ and then by $1$ . Then we subtract the second result from the first.

$9734 \times 99 = 9734 \times (100 - 1)$

$= (9734 \times 100) - (9734 \times 1)$

$= 973400 - 9734$

$\boxed{963666}$

Diagram 3

Step 6 — Multiplying by 999 (Part vi)

Let us find the product of $23478$ and $999$ . We write $999$ as $1000 - 1$ . We use the distributive property for subtraction. We multiply $23478$ by $1000$ and by $1$ . Then we subtract the second product from the first.

$23478 \times 999 = 23478 \times (1000 - 1)$

$= (23478 \times 1000) - (23478 \times 1)$

$= 23478000 - 23478$

$\boxed{23454522}$

Answer

(i) 8989 (ii) 95849 (iii) 266096831 (iv) 1112111 (v) 963666 (vi) 23454522

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