Compute these products using the suggested identity. (i) using Identity 1A for (ii) using Identity 1C for (iii) using Identity 1B for (iv) using Identity 1C for

Understand the Question Algebraic identities allow us to compute products and squares of numbers mentally and quickly by decomposing them relative to convenient base numbers (such as multiples of or ): Identity 1A: Identity 1B: Identity 1C: Part (i) using Identity 1A for Step 1 · Apply Identity 1A to calculate Write as , where and . Using Answer > (i) Part (ii) using Identity 1C for Step 1 · Apply Identity 1C to calculate Write and , where and . Using Answer > (ii) Part (iii) using Identity 1B for Step 1 · Apply Identity 1B to calculate Write as , where and . Using Answer > (iii) Part (iv) using Identity 1C for Step 1 · Apply Identity 1C to calculate The central number between and is . Write and , where and . Using Answer > (iv) Common Mistakes Missing the middle term: Mistakenly writing or without including the term. Sign error in Identity 1B: Writing instead of at the end. Choosing incorrect values for and : In , must be exactly midway between the two numbers so that the deviations and are identical.

Distributivity and Algebra | FIO

Question 10

Compute these products using the suggested identity.

(i) $46^2$ using Identity 1A for $(a + b)^2$

(ii) $397 \times 403$ using Identity 1C for $(a + b)(a - b)$

(iii) $91^2$ using Identity 1B for $(a - b)^2$

(iv) $43 \times 45$ using Identity 1C for $(a + b)(a - b)$

Solution

Understand the Question

Algebraic identities allow us to compute products and squares of numbers mentally and quickly by decomposing them relative to convenient base numbers (such as multiples of $10$ or $100$ ):

Identity 1A: $(a + b)^2 = a^2 + 2ab + b^2$
Identity 1B: $(a - b)^2 = a^2 - 2ab + b^2$
Identity 1C: $(a + b)(a - b) = a^2 - b^2$

(i) $46^2$ using Identity 1A for $(a + b)^2$

Step 1 · Apply Identity 1A to calculate $46^2$

Write $46$ as $(40 + 6)$ , where $a = 40$ and $b = 6$ . Diagram 1

Using $(a + b)^2 = a^2 + 2ab + b^2$

\begin{aligned} 46^2 &= (40 + 6)^2 \\ &= 40^2 + (2 \times 40 \times 6) + 6^2 \\ &= 1600 + 480 + 36 \\ &= 2080 + 36 \\ &= 2116 \end{aligned}

Answer

(i) $2116$

(ii) $397 \times 403$ using Identity 1C for $(a + b)(a - b)$

Step 1 · Apply Identity 1C to calculate $397 \times 403$

Write $397 = 400 - 3$ and $403 = 400 + 3$ , where $a = 400$ and $b = 3$ .

Using $(a - b)(a + b) = a^2 - b^2$

\begin{aligned} 397 \times 403 &= (400 - 3)(400 + 3) \\ &= 400^2 - 3^2 \\ &= 160000 - 9 \\ &= 159991 \end{aligned}

Answer

(ii) $159991$

(iii) $91^2$ using Identity 1B for $(a - b)^2$

Step 1 · Apply Identity 1B to calculate $91^2$

Write $91$ as $(100 - 9)$ , where $a = 100$ and $b = 9$ . Diagram 2

Using $(a - b)^2 = a^2 - 2ab + b^2$

\begin{aligned} 91^2 &= (100 - 9)^2 \\ &= 100^2 - (2 \times 100 \times 9) + 9^2 \\ &= 10000 - 1800 + 81 \\ &= 8200 + 81 \\ &= 8281 \end{aligned}

Answer

(iii) $8281$

(iv) $43 \times 45$ using Identity 1C for $(a + b)(a - b)$

Step 1 · Apply Identity 1C to calculate $43 \times 45$

The central number between $43$ and $45$ is $44$ . Write $43 = 44 - 1$ and $45 = 44 + 1$ , where $a = 44$ and $b = 1$ .

Using $(a - b)(a + b) = a^2 - b^2$

\begin{aligned} 43 \times 45 &= (44 - 1)(44 + 1) \\ &= 44^2 - 1^2 \\ &= 1936 - 1 \\ &= 1935 \end{aligned}

Answer

(iv) $1935$

Common Mistakes

Missing the middle term: Mistakenly writing $(a + b)^2 = a^2 + b^2$ or $(a - b)^2 = a^2 - b^2$ without including the $\pm 2ab$ term.
Sign error in Identity 1B: Writing $(a - b)^2 = a^2 - 2ab - b^2$ instead of $+ b^2$ at the end.
Choosing incorrect values for $a$ and $b$ : In $(a - b)(a + b)$ , $a$ must be exactly midway between the two numbers so that the deviations $+b$ and $-b$ are identical.

More questions in FIO

Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the middle number of a $3 \times 3$ frame is given by the expression $pq$ , as shown in the figure, write the expressions for the other numbers in the grid.

Expand the following products.

(i) $(3 + u) (v - 3)$

(ii) $\dfrac{2}{3} (15 + 6a)$

(iii) $(10a + b) (10c + d)$

(iv) $(3 - x) (x - 6)$

(v) $(-5a + b) (c + d)$

(vi) $(5 + z) (y + 9)$

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Expand:

(i) $(a + ab - 3b^2)(4 + b)$

(ii) $(4y + 7)(y + 11z - 3)$

Expand:

(i) $(a - b)(a + b)$

(ii) $(a - b)(a^2 + ab + b^2)$

(iii) $(a - b)(a^3 + a^2b + ab^2 + b^3)$

Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Which is greater: $(a - b)^2$ or $(b - a)^2$ ? Justify your answer.

Express 100 as the difference of two squares.

Find $406^2$ , $72^2$ , $145^2$ , $1097^2$ , and $124^2$ using the identities you have learnt so far.

Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.

Q10

Compute these products using the suggested identity.

(i) $46^2$ using Identity 1A for $(a + b)^2$

(ii) $397 \times 403$ using Identity 1C for $(a + b)(a - b)$

(iii) $91^2$ using Identity 1B for $(a - b)^2$

(iv) $43 \times 45$ using Identity 1C for $(a + b)(a - b)$

Q11

Use either a suitable identity or the distributive property to find each of the following products.

(i) $(p - 1)(p + 11)$

(ii) $(3a - 9b)(3a + 9b)$

(iii) $-(2y + 5)(3y + 4)$

(iv) $(6x + 5y)^2$

(v) $\left(2x - \dfrac{1}{2}\right)^2$

(vi) $(7p) \times (3r) \times (p + 2)$

Q12

For each statement identify the appropriate algebraic expression(s).

(i) Two more than a square number.

$2 + s \qquad (s + 2)^2 \qquad s^2 + 2 \qquad s^2 + 4 \qquad 2s^2 \qquad 2^2s$

(ii) The sum of the squares of two consecutive numbers

$m^2 + n^2 \qquad (m + n)^2 \qquad m^2 + 1 \qquad m^2 + (m + 1)^2$ $m^2 + (m - 1)^2 \qquad (m + (m + 1))^2 \qquad (2m)^2 + (2m + 1)^2$

Q13

Consider any 2 by 2 square of numbers in a calendar, as shown in the figure.

Find products of numbers lying along each diagonal — $4 \times 12 = 48$ , $5 \times 11 = 55$ . Do this for the other 2 by 2 squares. What do you observe about the diagonal products? Explain why this happens.

Hint: Label the numbers in each 2 by 2 square as shown in the diagram.

Q14

Verify which of the following statements are true.

(i) $(k + 1)(k + 2) - (k + 3)$ is always 2.

(ii) $(2q + 1)(2q - 3)$ is a multiple of 4.

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

(iv) $(6n + 2)^2 - (4n + 3)^2$ is 5 less than a square number.

Q15

A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?

Q16

Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same with other sets of numbers. What pattern do you notice? How do we write this as an algebraic equation? Expand both sides of the equation to check that it is a true identity.

Q17

What is the algebraic expression describing the following steps—add any two numbers. Multiply this by half of the sum of the two numbers? Prove that this result will be half of the square of the sum of the two numbers.

Q18

Which is larger? Find out without fully computing the product.

(i) $14 \times 26$ or $16 \times 24$

(ii) $25 \times 75$ or $26 \times 74$

Q19

A tiny park is coming up in Dhauli. The plan is shown in the figure. The two square plots, each of area $g^2 \text{ sq. ft.}$ , will have a green cover. All the remaining area is a walking path $w \text{ ft.}$ wide that needs to be tiled. Write an expression for the area that needs to be tiled.

Q20

For each pattern shown below,

(i) Draw the next figure in the sequence. (ii) How many basic units are there in Step 10? (iii) Write an expression to describe the number of basic units in Step $y$ .