Arithmetic Expressions | FIO

Question 18

Compare the following pairs of expressions using ' $<$ ', ' $>$ ' or ' $=$ ' or by reasoning.

(a) $49 - 7 + 8 \quad \square \quad 49 - 7 + 8$

(b) $83 \times 42 - 18 \quad \square \quad 83 \times 40 - 18$

(d) $23 \times 48 - 35 \quad \square \quad 23 \times (48 - 35)$

(e) $(16 - 11) \times 12 \quad \square \quad -11 \times 12 + 16 \times 12$

(f) $(76 - 53) \times 88 \quad \square \quad 88 \times (53 - 76)$

(g) $25 \times (42 + 16) \quad \square \quad 25 \times (43 + 15)$

(h) $36 \times (28 - 16) \quad \square \quad 35 \times (27 - 15)$

Solution

Understand the Question

Expressions can often be compared by reasoning (using properties of operations like the distributive and commutative properties, or comparing common terms) without doing long multiplications.
Key Rules for Comparison:
- Subtracting a larger number: If $A > B$ , then $N - A < N - B$ .
- Signs: Any positive value is always greater than any negative value.
- Distributive Property: $a \times (b - c) = a \times b - a \times c$ .

(a) $49 - 7 + 8 \quad \square \quad 49 - 7 + 8$

Step 1 · Compare Identical Expressions

Both expressions are identical on both sides: $49 - 7 + 8 = 49 - 7 + 8$

Answer

(a) $=$

(b) $83 \times 42 - 18 \quad \square \quad 83 \times 40 - 18$

Step 1 · Compare First Terms

Both expressions subtract $18$ . Comparing the first terms: $42 > 40 \implies 83 \times 42 > 83 \times 40$

Subtracting $18$ from both sides preserves the inequality: $83 \times 42 - 18 > 83 \times 40 - 18$

Answer

(b) $>$

(c) $145 - 17 \times 8 \quad \square \quad 145 - 17 \times 6$

Step 1 · Compare Subtracted Terms

Both expressions start from $145$ . Comparing the terms being subtracted: $8 > 6 \implies 17 \times 8 > 17 \times 6$

Subtracting a larger number from the same value gives a smaller result: $145 - 17 \times 8 < 145 - 17 \times 6$

Answer

(c) $<$

(d) $23 \times 48 - 35 \quad \square \quad 23 \times (48 - 35)$

Step 1 · Apply Distributive Property and Compare

Expanding the second expression using the distributive property: $23 \times (48 - 35) = 23 \times 48 - 23 \times 35$

Evaluating the subtracted term: $\begin{aligned} 23 \times 35 &= 805 \end{aligned}$

Comparing $23 \times 48 - 35$ with $23 \times 48 - 805$ : Since $35 < 805$ , the first expression subtracts a smaller number and is therefore larger: $23 \times 48 - 35 > 23 \times (48 - 35)$

Answer

(d) $>$

(e) $(16 - 11) \times 12 \quad \square \quad -11 \times 12 + 16 \times 12$

Step 1 · Apply Distributive and Commutative Properties

Expanding the first expression using the distributive property: $(16 - 11) \times 12 = 16 \times 12 - 11 \times 12$

By the commutative property of addition: $16 \times 12 - 11 \times 12 = -11 \times 12 + 16 \times 12$

Both expressions are equal: $(16 - 11) \times 12 = -11 \times 12 + 16 \times 12$

Answer

(e) $=$

(f) $(76 - 53) \times 88 \quad \square \quad 88 \times (53 - 76)$

Step 1 · Compare the Signs of Both Expressions

Evaluating the expressions inside the parentheses: $76 - 53 = 23 \implies (76 - 53) \times 88 = 23 \times 88 \quad (\text{positive})$

$53 - 76 = -23 \implies 88 \times (53 - 76) = 88 \times (-23) \quad (\text{negative})$

Since any positive number is greater than any negative number: $(76 - 53) \times 88 > 88 \times (53 - 76)$

Answer

(f) $>$

(g) $25 \times (42 + 16) \quad \square \quad 25 \times (43 + 15)$

Step 1 · Compare Sums Inside Parentheses

Evaluating the sum inside the parentheses for both expressions: $42 + 16 = 58 \implies 25 \times (42 + 16) = 25 \times 58$

$43 + 15 = 58 \implies 25 \times (43 + 15) = 25 \times 58$

Both expressions are equal: $25 \times (42 + 16) = 25 \times (43 + 15)$

Answer

(g) $=$

(h) $36 \times (28 - 16) \quad \square \quad 35 \times (27 - 15)$

Step 1 · Evaluate and Compare Both Expressions

Evaluating the first expression: $\begin{aligned} 36 \times (28 - 16) &= 36 \times 12 \\ &= 432 \end{aligned}$

Evaluating the second expression: $\begin{aligned} 35 \times (27 - 15) &= 35 \times 12 \\ &= 420 \end{aligned}$

Since $432 > 420$ : $36 \times (28 - 16) > 35 \times (27 - 15)$

Answer

(h) $>$

Common Mistakes

Parentheses vs. No Parentheses: In part (d), confusing $23 \times 48 - 35$ with $23 \times (48 - 35)$ . The second multiplies both $48$ and $35$ by $23$ .
Subtraction Sign Order: In part (f), assuming $76 - 53 = 53 - 76$ . Reversing the order of subtraction changes the sign of the result ( $23$ vs $-23$ ).
Subtraction Inequality Direction: Forgetting that subtracting a larger quantity produces a smaller overall value.

More questions in FIO

Fill in the blanks to make the expressions equal on both sides of the $=$ sign:

(a) $13 + 4 = \underline{\quad} + 6$

(b) $22 + \underline{\quad} = 6 \times 5$

(d) $34 - \underline{\quad} = 25$

Arrange the following expressions in ascending (increasing) order of their values.

(a) $67 - 19$

(b) $67 - 20$

(d) $5 \times 11$

(e) $120 \div 3$

Find the values of the following expressions by writing the terms in each case.

(a) $28 - 7 + 8$

(b) $39 - 2 \times 6 + 11$

(d) $48 - 10 \times 2 + 16 \div 2$

(e) $6 \times 3 - 4 \times 8 \times 5$

Write a story/situation for each of the following expressions and find their values.

(a) $89 + 21 - 10$

(b) $5 \times 12 - 6$

For each of the following situations, write the expression describing the situation, identify its terms and find the value of the expression.

(a) Queen Alia gave 100 gold coins to Princess Elsa and 100 gold coins to Princess Anna last year. Princess Elsa used the coins to start a business and doubled her coins. Princess Anna bought jewellery and has only half of the coins left. Write an expression describing how many gold coins Princess Elsa and Princess Anna together have.

(b) A metro train ticket between two stations is ₹40 for an adult and ₹20 for a child. What is the total cost of tickets:

(i) for four adults and three children?

(ii) for two groups having three adults each?

(c) Find the total height of the window by writing an expression describing the relationship among the measurements shown in the picture.

Add brackets at appropriate places in the expressions such that they lead to the values indicated.

(a) $34 - 9 + 12 = 13$

(b) $56 - 14 - 8 = 34$

Using only reasoning of how terms change their values, fill the blanks to make the expressions on either side of the equality ( $=$ ) equal.

(a) $423 + \underline{\quad\quad} = 419 + \underline{\quad\quad}$

(b) $207 - 68 = 210 - \underline{\quad\quad}$

Using the numbers 2, 3 and 5, and the operators '+' and '-', and brackets, as necessary, generate expressions to give as many different values as possible. For example, $2 - 3 + 5 = 4$ and $3 - (5 - 2) = 0$ .

Whenever Jasoda has to subtract 9 from a number, she subtracts 10 and adds 1 to it. For example, $36 - 9 = 26 + 1$ .

(a) Do you think she always gets the correct answer? Why?

(b) Can you think of other similar strategies? Give some examples.

Q10

Consider the two expressions: a) $73 - 14 + 1$ , b) $73 - 14 - 1$ . For each of these expressions, identify the expressions from the following collection that are equal to it.

(a) $73 - (14 + 1)$

(b) $73 - (14 - 1)$

(d) $73 + (-14 - 1)$

Q11

Figure it Out

Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal.

(a) $3 \times (6 + 7) = 3 \times 6 + 3 \times 7$

(b) $(8 + 3) \times 4 = 8 \times 4 + 3 \times 4$

(d) $(9 + 2) \times 4 = 9 \times 4 \ \Box\ 2 \times \underline{\ \ \ }$

(e) $3 \times (\underline{\ \ \ } + 4) = 3 \ \underline{\ \ \ } + \underline{\ \ \ }$

(f) $(\underline{\ \ \ } + 6) \times 4 = 13 \times 4 + \underline{\ \ \ }$

(g) $3 \times (\underline{\ \ \ } + \underline{\ \ \ }) = 3 \times 5 + 3 \times 2$

(h) $(\underline{\ \ \ } + \underline{\ \ \ }) \times \underline{\ \ \ } = 2 \times 4 + 3 \times 4$

(i) $5 \times (9 - 2) = 5 \times 9 - 5 \times \underline{\ \ \ }$

(j) $(5 - 2) \times 7 = 5 \times 7 - 2 \times \underline{\ \ \ }$

(k) $5 \times (8 - 3) = 5 \times 8 \ \Box\ 5 \times \underline{\ \ \ }$

(l) $(8 - 3) \times 7 = 8 \times 7 \ \Box\ 3 \times 7$

(m) $5 \times (12 - \underline{\ \ \ }) = \underline{\ \ \ } \ \Box\ 5 \times \underline{\ \ \ }$

(n) $(15 - \underline{\ \ \ }) \times 7 = \underline{\ \ \ } \ \Box\ 6 \times 7$

(o) $5 \times (\underline{\ \ \ } - \underline{\ \ \ }) = 5 \times 9 - 5 \times 4$

(p) $(\underline{\ \ \ } - \underline{\ \ \ }) \times \underline{\ \ \ } = 17 \times 7 - 9 \times 7$

Q12

In the boxes below, fill '<', '>' or '=' after analysing the expressions on the LHS and RHS. Use reasoning and understanding of terms and brackets to figure this out and not by evaluating the expressions.

(a) $(8 - 3) \times 29 \ \square\ (3 - 8) \times 29$

(b) $15 + 9 \times 18 \ \square\ (15 + 9) \times 18$

(d) $(34 - 28) \times 42 \ \square\ 34 \times 42 - 28 \times 42$

Q13

Here is one way to make 14: $2 \times (1 + 6) = 14$ . Are there other ways of getting 14? Fill them out below:

(a) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

(b) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

(d) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

Q14

Find out the sum of the numbers given in each picture below in at least two different ways. Describe how you solved it through expressions.

Q15

Read the situations given below. Write appropriate expressions for each of them and find their values.

(a) The district market in Begur operates on all seven days of a week. Rahim supplies 9 kg of mangoes each day from his orchard and Shyam supplies 11 kg of mangoes each day from his orchard to this market. Find the amount of mangoes supplied by them in a week to the local district market.

(b) Binu earns ₹20,000 per month. She spends ₹5,000 on rent, ₹5,000 on food, and ₹2,000 on other expenses every month. What is the amount Binu will save by the end of a year?

(c) During the daytime a snail climbs 3 cm up a post, and during the night while asleep, accidentally slips down by 2 cm. The post is 10 cm high, and a delicious treat is on its top. In how many days will the snail get the treat?

Q16

Melvin reads a two-page story every day except on Tuesdays and Saturdays. How many stories would he complete reading in 8 weeks? Which of the expressions below describes this scenario?

(a) $5 \times 2 \times 8$

(b) $(7 - 2) \times 8$

(d) $7 \times 2 \times 8$

(e) $7 \times 5 - 2$

(f) $(7 + 2) \times 8$

(g) $7 \times 8 - 2 \times 8$

(h) $(7 - 5) \times 8$

Q17

Find different ways of evaluating the following expressions:

(a) $1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10$

(b) $1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

Q18

Compare the following pairs of expressions using ' $<$ ', ' $>$ ' or ' $=$ ' or by reasoning.

(a) $49 - 7 + 8 \quad \square \quad 49 - 7 + 8$

(b) $83 \times 42 - 18 \quad \square \quad 83 \times 40 - 18$

(d) $23 \times 48 - 35 \quad \square \quad 23 \times (48 - 35)$

(e) $(16 - 11) \times 12 \quad \square \quad -11 \times 12 + 16 \times 12$

(f) $(76 - 53) \times 88 \quad \square \quad 88 \times (53 - 76)$

(g) $25 \times (42 + 16) \quad \square \quad 25 \times (43 + 15)$

(h) $36 \times (28 - 16) \quad \square \quad 35 \times (27 - 15)$

Q19

Identify which of the following expressions are equal to the given expression without computation. You may rewrite the expressions using terms or removing brackets. There can be more than one expression which is equal to the given expression.

(a) $83 - 37 - 12$

(i) $84 - 38 - 12$

(ii) $84 - (37 + 12)$

(iii) $83 - 38 - 13$

(iv) $-37 + 83 - 12$

(b) $93 + 37 \times 44 + 76$

(i) $37 + 93 \times 44 + 76$

(ii) $93 + 37 \times 76 + 44$

(iii) $(93 + 37) \times (44 + 76)$

(iv) $37 \times 44 + 93 + 76$

Q20

Choose a number and create ten different expressions having that value.