Mind the Mistake, Mend the Mistake Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form. Observe each of them and see if there is a mistake. If you think there is a mistake, try to explain what might have gone wrong. Then, simplify it correctly.

Expressions Using Letter-Numbers | IT

Question 19

Mind the Mistake, Mend the Mistake

Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

Observe each of them and see if there is a mistake.
If you think there is a mistake, try to explain what might have gone wrong.
Then, simplify it correctly.

Solution

We will check each algebraic simplification for mistakes. We will explain any errors and then provide the correct simplest form.

Step 1 — Identify Mistake 1

Let us look at the first expression. The expression is $3a + 2b$ . The given simplest form is $\textbf{5}$ . There is a mistake here. The student made an error.

Step 2 — Explain Mistake 1

We cannot add terms with different variables. The term $3a$ has variable $a$ . The term $2b$ has variable $b$ . These are called unlike terms. We can only add or subtract like terms.

Step 3 — Correct Simplification 1

Let us simplify the expression correctly. The expression is $3a + 2b$ . It has unlike terms. So, we cannot combine them. The expression is already in its simplest form.

$\boxed{\textbf{3a + 2b}}$

Step 4 — Check Simplification 2

Let us look at the second expression. The expression is $3b - 2b - b$ . The given simplest form is $\textbf{0}$ . This simplification is correct. We combine the like terms. $3b - 2b - b$ $= (3 - 2 - 1)b$ $= 0b$

$\boxed{\textbf{0}}$

Step 5 — Identify Mistake 3

Let us look at the third expression. The expression is $6(p + 2)$ . The given simplest form is $6p + 8$ . There is a mistake here.

Step 6 — Explain Mistake 3

We use the distributive property. The number outside multiplies each term inside. The number $\textbf{6}$ must multiply $\textbf{p}$ . The number $\textbf{6}$ must also multiply $\textbf{2}$ . The student forgot to multiply $\textbf{6}$ by $\textbf{2}$ . They wrote $\textbf{8}$ instead of $\textbf{12}$ .

Step 7 — Correct Simplification 3

Let us simplify the expression correctly. We apply the distributive property. $6(p + 2)$ $= (6 \times p) + (6 \times 2)$ $= 6p + 12$

$\boxed{\textbf{6p + 12}}$

Step 8 — Identify Mistake 4

Let us look at the fourth expression. The expression is $(4x + 3y) - (3x + 4y)$ . The given simplest form is $x + y$ . There is a mistake here.

Step 9 — Explain Mistake 4

When subtracting an expression in parentheses, we change signs. We change the sign of each term inside. The student did not change the sign of $\textbf{4y}$ . It should be $\textbf{-4y}$ , not $\textbf{+4y}$ .

Step 10 — Correct Simplification 4

Let us simplify the expression correctly. We remove the parentheses carefully. $(4x + 3y) - (3x + 4y)$ $= 4x + 3y - 3x - 4y$ $= (4x - 3x) + (3y - 4y)$ $= x - y$

$\boxed{\textbf{x - y}}$

Step 11 — Identify Mistake 5

Let us look at the fifth expression. The expression is $5 - (2 - 6z)$ . The given simplest form is $3 - 6z$ . There is a mistake here.

Step 12 — Explain Mistake 5

When subtracting an expression in parentheses, we change signs. We change the sign of each term inside. The student did not change the sign of $\textbf{-6z}$ . It should be $\textbf{+6z}$ , not $\textbf{-6z}$ .

Step 13 — Correct Simplification 5

Let us simplify the expression correctly. We remove the parentheses carefully. $5 - (2 - 6z)$ $= 5 - 2 + 6z$ $= 3 + 6z$

$\boxed{\textbf{3 + 6z}}$

Step 14 — Identify Mistake 6

Let us look at the sixth expression. The expression is $2 + (x + 3)$ . The given simplest form is $2x - 6$ . There is a mistake here.

Step 15 — Explain Mistake 6

There is no multiplication in this expression. The student incorrectly multiplied $\textbf{2}$ by $\textbf{x}$ and $\textbf{3}$ . They also changed the signs. We just remove the parentheses.

Step 16 — Correct Simplification 6

Let us simplify the expression correctly. We remove the parentheses. $2 + (x + 3)$ $= 2 + x + 3$ $= x + 5$

$\boxed{\textbf{x + 5}}$

Step 17 — Identify Mistake 7

Let us look at the seventh expression. The expression is $2y + (3y - 6)$ . The given simplest form is $-y + 6$ . There is a mistake here.

Step 18 — Explain Mistake 7

We just remove the parentheses. Then we combine the like terms. The student incorrectly combined the $\textbf{y}$ terms. They also changed the sign of $\textbf{6}$ .

Step 19 — Correct Simplification 7

Let us simplify the expression correctly. We remove the parentheses. $2y + (3y - 6)$ $= 2y + 3y - 6$ $= 5y - 6$

$\boxed{\textbf{5y - 6}}$

Step 20 — Identify Mistake 8

Let us look at the eighth expression. The expression is $7p - p + 5q - 2q$ . The given simplest form is $7p + 3q$ . There is a mistake here.

Step 21 — Explain Mistake 8

We combine the like terms. The term $\textbf{p}$ means $\textbf{1p}$ . The student incorrectly combined $\textbf{7p - p}$ . It should be $\textbf{6p}$ , not $\textbf{7p}$ .

Step 22 — Correct Simplification 8

Let us simplify the expression correctly. We group the like terms. $7p - p + 5q - 2q$ $= (7p - 1p) + (5q - 2q)$ $= 6p + 3q$

$\boxed{\textbf{6p + 3q}}$

Step 23 — Identify Mistake 9

Let us look at the ninth expression. The expression is $5(2w + 3x + 4w)$ . The given simplest form is $10w + 15x + 20w$ . There is a mistake here.

Step 24 — Explain Mistake 9

The expression is not in its simplest form. The terms $\textbf{10w}$ and $\textbf{20w}$ are like terms. They can still be combined. The student did not combine all like terms.

Step 25 — Correct Simplification 9

Let us simplify the expression correctly. First, combine like terms inside parentheses. $5(2w + 3x + 4w)$ $= 5(6w + 3x)$ Now, apply the distributive property. $= (5 \times 6w) + (5 \times 3x)$ $= 30w + 15x$

$\boxed{\textbf{30w + 15x}}$

Step 26 — Check Simplification 10

Let us look at the tenth expression. The expression is $3j + 6k + 9h + 12$ . The given simplest form is $3(j + 2k + 3h + 4)$ . This factorization is correct. We can factor out the common factor. The common factor for all terms is $\textbf{3}$ . $3j + 6k + 9h + 12$ $= 3(j + 2k + 3h + 4)$

$\boxed{\textbf{3(j + 2k + 3h + 4)}}$

Step 27 — Identify Mistake 11

Let us look at the eleventh expression. The expression is $4(2r + 3s + 5)$ . The given simplest form is $-20 - 8r - 12s$ . There is a mistake here.

Step 28 — Explain Mistake 11

We use the distributive property. The number outside multiplies each term inside. The number $\textbf{4}$ is positive. So, all terms inside should remain positive. The student incorrectly changed all the signs.

Step 29 — Correct Simplification 11

Let us simplify the expression correctly. We apply the distributive property. $4(2r + 3s + 5)$ $= (4 \times 2r) + (4 \times 3s) + (4 \times 5)$ $= 8r + 12s + 20$

$\boxed{\textbf{8r + 12s + 20}}$

Answer

(i) Correct Simplest Form for 1: 3a + 2b (ii) Correct Simplest Form for 2: 0 (iii) Correct Simplest Form for 3: 6p + 12 (iv) Correct Simplest Form for 4: x - y (v) Correct Simplest Form for 5: 3 + 6z (vi) Correct Simplest Form for 6: x + 5 (vii) Correct Simplest Form for 7: 5y - 6 (viii) Correct Simplest Form for 8: 6p + 3q (ix) Correct Simplest Form for 9: 30w + 15x (x) Correct Simplest Form for 10: 3(j + 2k + 3h + 4) (xi) Correct Simplest Form for 11: 8r + 12s + 20

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Mind the Mistake, Mend the Mistake

Some simplifications are shown below where the letter-numbers are replaced by numbers and the value of the expression is obtained.

Observe each of them and identify if there is a mistake.
If you think there is a mistake, try to explain what might have gone wrong.
Then, correct it and give the value of the expression.

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The total money earned by the sale of pencils is $5c + 3c + 10c = 18c$ .

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Context: Example 5: Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is $c$ , and the price per eraser is $d$ . Find the total money earned by the shopkeeper during these three days.

Q. Write the expression for the total money earned by selling erasers. Then, simplify the expression.

Q10

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Q. Check that both expressions take the same value when $c$ is replaced by different numbers.

Q11

Context: The expression $(40x + 75y) - (6x + 10y)$ is simplified to $34x + 65y$ , which is the total amount paid in rupees.

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Q12

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Q13

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Q14

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Q. Give some possible scores for Krishita in the three rounds so that they add up to give $23p - 7q$ .

Q15

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Q16

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Q. Simplify this expression further.

Q17

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Q18

Context: Let us compare the values that the expressions $10y - 3$ and $10(y - 3)$ take for different values of $y$ .

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Q19

Mind the Mistake, Mend the Mistake

Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

Observe each of them and see if there is a mistake.
If you think there is a mistake, try to explain what might have gone wrong.
Then, simplify it correctly.

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Q21

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Q22

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Q23

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Q24

Context: Let us take the marked $2 \times 2$ square, and consider the numbers lying on the diagonals; 12 and 20; 13 and 19. Find their sums; $12 + 20$ , $13 + 19$ . What do you observe? They are equal. Let us extend the numbers in the calendar beyond 30, creating endless rows.

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Q26

Verify this expression for diagonal sums by considering any $2 \times 2$ square and taking its top left number to be 'a'.

Q27

Context: Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

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Q30

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Q. What are these numbers in Step 3 and Step 4?

Q34

Q. How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step ‘y’ in each orientation. Do the two expressions add up to 2y + 1?

Q35

Numbers are written in a particular sequence in this endless 4-column grid.

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201

(d) Observe the positions of multiples of 3.

Do you see any pattern in it? List other patterns that you see.