Arithmetic Expressions | FIO

Question 19

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) $37 \times 44 + 93 + 76$

Solution

We will use the rules for order of operations and properties of numbers.

Step 1 — Analyze expression (a)

The given expression is $83 - 37 - 12$ . We can rewrite this as $83 + (-37) + (-12)$ . This helps us see the terms clearly.

Let us check option (i). The expression is $84 - 38 - 12$ . We can write $84$ as $83 + 1$ . We can write $38$ as $37 + 1$ . Let us substitute these values. $84 - 38 - 12$ $= (83 + 1) - (37 + 1) - 12$ $= 83 + 1 - 37 - 1 - 12$ We can rearrange the terms. $= 83 - 37 - 12 + 1 - 1$ $= 83 - 37 - 12 + 0$ This is the same as the given expression. So, option (i) is equal.

Let us check option (ii). The expression is $84 - (37 + 12)$ . The given expression is $83 - 37 - 12$ . We know that $A - B - C = A - (B + C)$ . So, $83 - 37 - 12 = 83 - (37 + 12)$ . Comparing $84 - (37 + 12)$ with $83 - (37 + 12)$ . The first number is different ( $84$ instead of $83$ ). So, option (ii) is not equal.

Let us check option (iii). The expression is $83 - 38 - 13$ . The given expression is $83 - 37 - 12$ . The first number is the same. The second number changed from $37$ to $38$ . The third number changed from $12$ to $13$ . Let us rewrite the option. $83 - 38 - 13$ $= 83 - (37 + 1) - (12 + 1)$ $= 83 - 37 - 1 - 12 - 1$ $= 83 - 37 - 12 - 1 - 1$ $= 83 - 37 - 12 - 2$ This is not the same as the given expression. So, option (iii) is not equal.

Let us check option (iv). The expression is $- 37 + 83 - 12$ . The given expression is $83 - 37 - 12$ . We can write $83 - 37 - 12$ as $83 + (-37) + (-12)$ . Addition allows us to change the order of terms. So, $83 + (-37) + (-12)$ is the same as $(-37) + 83 + (-12)$ . This is $-37 + 83 - 12$ . So, option (iv) is equal.

Diagram 1

Step 2 — Analyze expression (b)

The given expression is $93 + 37 \times 44 + 76$ . We must follow the order of operations. Multiplication comes before addition. So, we calculate $37 \times 44$ first. The expression is $93 + (37 \times 44) + 76$ .

Let us check option (i). The expression is $37 + 93 \times 44 + 76$ . Following the order of operations, this is $37 + (93 \times 44) + 76$ . The terms being added are $37$ , $(93 \times 44)$ , and $76$ . In the given expression, the terms are $93$ , $(37 \times 44)$ , and $76$ . These sets of terms are different. So, option (i) is not equal.

Let us check option (ii). The expression is $93 + 37 \times 76 + 44$ . Following the order of operations, this is $93 + (37 \times 76) + 44$ . The multiplication part is $37 \times 76$ . In the given expression, the multiplication part is $37 \times 44$ . These are different. So, option (ii) is not equal.

Let us check option (iii). The expression is $(93 + 37) \times (44 + 76)$ . This expression involves adding numbers first, then multiplying the sums. The given expression involves multiplication first, then addition. The order of operations is completely changed. So, option (iii) is not equal.

Let us check option (iv). The expression is $37 \times 44 + 93 + 76$ . Following the order of operations, this is $(37 \times 44) + 93 + 76$ . The terms being added are $(37 \times 44)$ , $93$ , and $76$ . In the given expression, the terms are $93$ , $(37 \times 44)$ , and $76$ . Addition is commutative. This means we can change the order of terms being added. So, $93 + (37 \times 44) + 76$ is the same as $(37 \times 44) + 93 + 76$ . So, option (iv) is equal.

Answer

(a) (i) and (iv) are equal to the given expression. (b) (iv) is equal to the given expression.

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 × 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