Distributivity and Algebra | FIO

Question 18

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

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

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

Solution

Understand the Question

We can compare products of pairs with the same sum by rewriting them around their common midpoint (average) using the identity: $(a - b)(a + b) = a^2 - b^2$
Since both expressions share the same base term $a^2$ , the product where a smaller square $b^2$ is subtracted will have the larger value.

(i) Which is larger: $14 \times 26$ or $16 \times 24$ ?

Step 1 · Compare Using the Difference of Squares Identity

The midpoint of each pair is $20$ . Diagram 1

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

\begin{aligned} 14 \times 26 &= (20 - 6)(20 + 6) = 20^2 - 6^2 \\ 16 \times 24 &= (20 - 4)(20 + 4) = 20^2 - 4^2 \end{aligned}

Comparing the subtracted squares

\begin{aligned} 6^2 &= 36 \\ 4^2 &= 16 \end{aligned}

Since $6^2 > 4^2$ , subtracting $6^2$ gives a smaller result $20^2 - 6^2 < 20^2 - 4^2 \implies 14 \times 26 < 16 \times 24$

Answer

(i) $16 \times 24$ is larger

(ii) Which is larger: $25 \times 75$ or $26 \times 74$ ?

Step 1 · Compare Using the Difference of Squares Identity

The midpoint of each pair is $50$ .

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

\begin{aligned} 25 \times 75 &= (50 - 25)(50 + 25) = 50^2 - 25^2 \\ 26 \times 74 &= (50 - 24)(50 + 24) = 50^2 - 24^2 \end{aligned}

Comparing the subtracted squares

\begin{aligned} 25^2 &= 625 \\ 24^2 &= 576 \end{aligned}

Since $25^2 > 24^2$ , subtracting $25^2$ gives a smaller result $50^2 - 25^2 < 50^2 - 24^2 \implies 25 \times 75 < 26 \times 74$

Answer

(ii) $26 \times 74$ is larger

Common Mistakes

Subtraction Inversion Error: Forgetting that subtracting a larger square ( $b^2$ ) makes the total value smaller, not larger ( $a^2 - 6^2 < a^2 - 4^2$ ).
Incorrect Midpoint: Choosing an incorrect base value $a$ instead of the exact mean of the two numbers.

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 $y$ .