Arithmetic Expressions | FIO

Question 17

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

Solution

We will evaluate each expression in different ways.

Step 1 — Evaluating Expression (a) by Grouping Pairs

Let us look at the expression for part (a). We can group the numbers into pairs. Each pair consists of a positive and a negative number.

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

Let us group the terms like this:

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

Each group evaluates to -1. There are five such groups in total. So, we multiply the number of groups by their value.

$5 \times (-1)$

$= \mathbf{-5}$

$\boxed{\mathbf{-5}}$

Step 2 — Evaluating Expression (a) by Separating Positive and Negative Numbers

Let us try another way for part (a). We can separate all positive numbers. Then we separate all negative numbers.

The positive numbers are 1, 3, 5, 7, 9. Let us add these positive numbers.

$1 + 3 + 5 + 7 + 9$

$= 4 + 5 + 7 + 9$

$= 9 + 7 + 9$

$= 16 + 9$

$= \mathbf{25}$

The negative numbers are -2, -4, -6, -8, -10. We can write their sum as a negative of positive numbers.

$-(2 + 4 + 6 + 8 + 10)$

Let us add these positive numbers inside the bracket.

$2 + 4 + 6 + 8 + 10$

$= 6 + 6 + 8 + 10$

$= 12 + 8 + 10$

$= 20 + 10$

$= \mathbf{30}$

So, the sum of negative numbers is -30. Now, we add the positive sum and the negative sum.

$25 + (-30)$

$= 25 - 30$

$= \mathbf{-5}$

$\boxed{\mathbf{-5}}$

Step 3 — Evaluating Expression (b) by Grouping Pairs

Let us look at the expression for part (b). We can group the numbers into pairs. Each pair consists of a positive and a negative one.

$1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

Let us group the terms like this:

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

Each group evaluates to 0. There are five such groups in total. So, we multiply the number of groups by their value.

$5 \times 0$

$= \mathbf{0}$

$\boxed{\mathbf{0}}$

Step 4 — Evaluating Expression (b) by Separating Positive and Negative Numbers

Let us try another way for part (b). We can separate all positive numbers. Then we separate all negative numbers.

The positive numbers are 1, 1, 1, 1, 1. Let us add these positive numbers.

$1 + 1 + 1 + 1 + 1$

$= \mathbf{5}$

The negative numbers are -1, -1, -1, -1, -1. We can write their sum as a negative of positive ones.

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

Let us add these positive numbers inside the bracket.

$1 + 1 + 1 + 1 + 1$

$= \mathbf{5}$

So, the sum of negative numbers is -5. Now, we add the positive sum and the negative sum.

$5 + (-5)$

$= 5 - 5$

$= \mathbf{0}$

$\boxed{\mathbf{0}}$

Step 5 — Evaluating Expression (b) by Sequential Calculation

Let us try a third way for part (b). We can calculate the expression from left to right. We will perform each operation one by one.

$1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

$= 0 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

$= 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

$= 0 + 1 - 1 + 1 - 1 + 1 - 1$

$= 1 - 1 + 1 - 1 + 1 - 1$

$= 0 + 1 - 1 + 1 - 1$

$= 1 - 1 + 1 - 1$

$= 0 + 1 - 1$

$= 1 - 1$

$= \mathbf{0}$

$\boxed{\mathbf{0}}$

Answer

(a) The value of the expression is -5. (b) The value of the expression is 0.

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

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.