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

We will use a special algebraic identity to compare the products.

Step 1 — Comparing $14 \times 26$ and $16 \times 24$

An algebraic identity is an equation always true. We use the identity $(a-b)(a+b) = a^2 - b^2$ . This identity helps us compare products easily.

Let us look at the first product. We can write $14$ as $20 - 6$ . We can write $26$ as $20 + 6$ . So, $14 \times 26 = (20 - 6)(20 + 6)$ . Using the identity, this product is $20^2 - 6^2$ .

Now, let us look at the second product. We can write $16$ as $20 - 4$ . We can write $24$ as $20 + 4$ . So, $16 \times 24 = (20 - 4)(20 + 4)$ . Using the identity, this product is $20^2 - 4^2$ .

We need to compare $20^2 - 6^2$ and $20^2 - 4^2$ . Both expressions start with $20^2$ . We are subtracting $6^2$ from $20^2$ . We are subtracting $4^2$ from $20^2$ . Let us compare $6^2$ and $4^2$ . $6^2 = 36$ $4^2 = 16$ We see that $\textbf{36}$ is larger than $\textbf{16}$ . So, $6^2$ is larger than $4^2$ . When we subtract a larger number, the result is smaller. Therefore, $20^2 - 6^2$ is smaller than $20^2 - 4^2$ . This means $14 \times 26$ is smaller than $16 \times 24$ .

$\boxed{\textbf{16 \times 24 is larger}}$

Diagram 1

Step 2 — Comparing $25 \times 75$ and $26 \times 74$

We will use the same identity: $(a-b)(a+b) = a^2 - b^2$ .

Let us look at the first product. We can write $25$ as $50 - 25$ . We can write $75$ as $50 + 25$ . So, $25 \times 75 = (50 - 25)(50 + 25)$ . Using the identity, this product is $50^2 - 25^2$ .

Now, let us look at the second product. We can write $26$ as $50 - 24$ . We can write $74$ as $50 + 24$ . So, $26 \times 74 = (50 - 24)(50 + 24)$ . Using the identity, this product is $50^2 - 24^2$ .

We need to compare $50^2 - 25^2$ and $50^2 - 24^2$ . Both expressions start with $50^2$ . We are subtracting $25^2$ from $50^2$ . We are subtracting $24^2$ from $50^2$ . Let us compare $25^2$ and $24^2$ . $25^2 = 625$ $24^2 = 576$ We see that $\textbf{625}$ is larger than $\textbf{576}$ . So, $25^2$ is larger than $24^2$ . When we subtract a larger number, the result is smaller. Therefore, $50^2 - 25^2$ is smaller than $50^2 - 24^2$ . This means $25 \times 75$ is smaller than $26 \times 74$ .

$\boxed{\textbf{26 \times 74 is larger}}$

Answer

(i) $16 \times 24$ is larger. (ii) $26 \times 74$ is larger.

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

(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) $(2x - \frac{1}{2})^2$

(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 $g^2$ sq. ft., will have a green cover. All the remaining area is a walking path $w$ 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.