Arithmetic Expressions | FIO

Question 12

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

Solution

We will compare the expressions without calculating their exact values.

Step 1 — Comparing positive and negative products for (a)

Let us look at the first expression. The Left Hand Side (LHS) is $(8 - 3) \times 29$ . We calculate the value inside the bracket first. $8 - 3 = 5$ So, the LHS becomes $5 \times 29$ . This is a positive number multiplied by a positive number. The Right Hand Side (RHS) is $(3 - 8) \times 29$ . We calculate the value inside the bracket first. $3 - 8 = -5$ So, the RHS becomes $-5 \times 29$ . This is a negative number multiplied by a positive number. A positive product is always greater than a negative product. So, $5 \times 29$ is greater than $-5 \times 29$ .

$\boxed{(8 - 3) \times 29 \ > \ (3 - 8) \times 29}$

Step 2 — Understanding order of operations for (b)

Let us look at the second expression. The Left Hand Side (LHS) is $15 + 9 \times 18$ . We follow the order of operations (BODMAS/PEMDAS). Multiplication comes before addition. So, we first calculate $9 \times 18$ . The expression is $15 + (9 \times 18)$ . The Right Hand Side (RHS) is $(15 + 9) \times 18$ . We calculate the value inside the bracket first. $15 + 9 = 24$ So, the RHS becomes $24 \times 18$ . We can write $24 \times 18$ using the distributive property. $(15 + 9) \times 18 = 15 \times 18 + 9 \times 18$ Now we compare $15 + 9 \times 18$ with $15 \times 18 + 9 \times 18$ . Both expressions have the term $9 \times 18$ . We need to compare $15$ with $15 \times 18$ . We know that $15 \times 18$ is much larger than $15$ . So, $15 + 9 \times 18$ is less than $15 \times 18 + 9 \times 18$ .

$\boxed{15 + 9 \times 18 \ < \ (15 + 9) \times 18}$

Step 3 — Applying the distributive property for (c)

Let us look at the third expression. The Left Hand Side (LHS) is $23 \times (17 - 9)$ . We use the distributive property here. The property states $a \times (b - c) = a \times b - a \times c$ . So, $23 \times (17 - 9)$ becomes $23 \times 17 - 23 \times 9$ . The Right Hand Side (RHS) is $23 \times 17 + 23 \times 9$ . Now we compare $23 \times 17 - 23 \times 9$ with $23 \times 17 + 23 \times 9$ . Both expressions have the term $23 \times 17$ . On the LHS, we subtract $23 \times 9$ . On the RHS, we add $23 \times 9$ . Subtracting a positive number makes the value smaller. Adding a positive number makes the value larger. So, $23 \times 17 - 23 \times 9$ is less than $23 \times 17 + 23 \times 9$ .

$\boxed{23 \times (17 - 9) \ < \ 23 \times 17 + 23 \times 9}$

Step 4 — Recognizing the distributive property for (d)

Let us look at the fourth expression. The Left Hand Side (LHS) is $(34 - 28) \times 42$ . This is in the form $(a - b) \times c$ . The Right Hand Side (RHS) is $34 \times 42 - 28 \times 42$ . This is in the form $a \times c - b \times c$ . The distributive property states that $(a - b) \times c = a \times c - b \times c$ . The LHS and RHS are exactly the same by this property. So, the two expressions are equal.

$\boxed{(34 - 28) \times 42 \ = \ 34 \times 42 - 28 \times 42}$

Answer

(a) $(8 - 3) \times 29 \ \mathbf{>} \ (3 - 8) \times 29$ (b) $15 + 9 \times 18 \ \mathbf{<} \ (15 + 9) \times 18$ (c) $23 \times (17 - 9) \ \mathbf{<} \ 23 \times 17 + 23 \times 9$ (d) $(34 - 28) \times 42 \ \mathbf{=} \ 34 \times 42 - 28 \times 42$

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

(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 × 2 × 8

(b) (7 - 2) × 8

(d) 7 × 2 × 8

(e) 7 × 5 - 2

(f) (7 + 2) × 8

(g) 7 × 8 - 2 × 8

(h) (7 - 5) × 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.