Chapter 2 Arithmetic Expressions (Class 7 - Latest Maths NCERT (Ganita Prakash I) Solutions)

Question. Different expressions can have the same value. For example, here are multiple ways to express the number $12$, using two numbers and any of the four operations $+$, $-$, $\times$ and $\div$:

$10 + 2, 15 - 3, 3 \times 4, 24 \div 2$

Choose your favourite number and write as many expressions as you can having that value.

Question 1. Fill in the blanks to make the expressions equal on both sides of the $=$ sign:

(a) $13 + 4 = \text{____} + 6$

(b) $22 + \text{____} = 6 \times 5$

(c) $8 \times \text{____} = 64 \div 2$

(d) $34 - \text{____} = 25$

Question 2. Arrange the following expressions in ascending (increasing) order of their values.

(a) $67 - 19$

(b) $67 - 20$

(c) $35 + 25$

(d) $5 \times 11$

(e) $120 \div 3$

Question. Use ‘$>$’ or ‘$<$’ or ‘$=$’ in each of the following expressions to compare them. Can you do it without complicated calculations? Explain your thinking in each case.

(a) $245 + 289$ ________ $246 + 285$

(b) $273 - 145$ ________ $272 - 144$

(c) $364 + 587$ ________ $363 + 589$

(d) $124 + 245$ ________ $129 + 245$

(e) $213 - 77$ ________ $214 - 76$

Question. The inverse of $14$ is $-14$, and the inverse of $-14$ is $14$. Thus, subtracting $14$ from $83$ is the same as adding $-14$ to $83$. That is,

$83 - 14 = 83 + (-14)$

Thus, the terms of the expression $83 - 14$ are $83$ and $-14$. Check if replacing subtraction by addition in this way does not change the value of the expression, by taking different examples.

Question. Can you explain why subtracting a number is the same as adding its inverse, using the Token Model of integers that we saw in the Class $6$ textbook of mathematics?

Question. Complete the table.

Question. Does changing the order in which the terms are added give different values?

Question. Madhu is flying a drone from a terrace. The drone goes $6\text{ m}$ up and then $4\text{ m}$ down.

The drone is $6 - 4 = 2\text{ m}$ above the terrace. Writing it as sum of terms: $6 + (-4) = 2$.

Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.

Question. Can you explain why this is happening using the Token Model of integers that we saw in the Class $6$ textbook of mathematics?

Question. Now consider an expression having three terms: $(-7) + 10 + (-11)$. Let us add these terms in the following two different orders:

$-7 + 10 + (-11)$ (adding the first two terms and then adding their sum to the third term)

$-7 + 10 + (-11)$ (adding the last two terms and then adding their sum to the first term)

What do you see? The sums are the same in both cases. Again, we know that while adding positive numbers, grouping them in any of the above two ways gives the same sum. Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.

Question. Can you explain why this is happening using the Token Model of integers that we saw in the Class $6$ textbook of mathematics?

Question. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than $3$ terms also.

Question. Can you explain why this is happening using the Token Model of integers that we saw in the Class $6$ textbook of mathematics?

Question. Manasa is adding a long list of numbers. It took her five minutes to add them all and she got the answer $11749$. Then she realised that she had forgotten to include the fourth number $9055$. Does she have to start all over again?

$$ \begin{array}{r} 1342 \\ 774 \\ 8611 \\ 9055 \\ + 1022 \\ \hline \end{array} $$

Question. Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa cost $\text{₹}23$, and they wish to thank the waiter by tipping $\text{₹}5$. Write an expression describing the total cost.

Question. If the total number of friends goes up to $7$ and the tip remains the same, how much will they have to pay? Write an expression for this situation and identify its terms.

Question. Children in a class are playing “Fire in the mountain, run, run, run!”. Whenever the teacher calls out a number, students are supposed to arrange themselves in groups of that number. Whoever is not part of the announced group size, is out.

For each of the cases below, write the expression and identify its terms:

If the teacher had called out ‘$4$’, Ruby would write ____________

If the teacher had called out ‘$7$’, Ruby would write ____________

Write expressions like the above for your class size.

Question. Kannan has to pay $\text{₹}432$ to a shopkeeper using coins of $\text{₹}1$ and $\text{₹}5$, and notes of $\text{₹}10, \text{₹}20, \text{₹}50$ and $\text{₹}100$. How can he do it?

There is more than one possibility. For example,

$432 = 4 \times 100 + 1 \times 20 + 1 \times 10 + 2 \times 1$ (Meaning: $4$ notes of $\text{₹}100$, $1$ note of $\text{₹}20$, $1$ note of $\text{₹}10$ and $2$ notes of $\text{₹}1$)

$432 = 8 \times 50 + 1 \times 10 + 4 \times 5 + 2 \times 1$ (Meaning: $8$ notes of $\text{₹}50$, $1$ note of $\text{₹}10$, $4$ notes of $\text{₹}5$ and $2$ notes of $\text{₹}1$)

Identify the terms in the two expressions above. Can you think of some more ways of giving $\text{₹}432$ to someone?

Question 1. Find the values of the following expressions by writing the terms in each case.

(a) $28 - 7 + 8$

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

(c) $40 - 10 + 10 + 10$

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

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

Question 2. Write a story/situation for each of the following expressions and find their values.

(a) $89 + 21 - 10$

(b) $5 \times 12 - 6$

(c) $4 \times 9 + 2 \times 6$

Question 3. 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 $\text{₹}40$ for an adult and $\text{₹}20$ 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.

Question. Some expressions are given in following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Question 1. Fill in the blanks with numbers, and boxes with operation signs such that the expressions on both sides are equal.

(a) $24 + (6 - 4) = 24 + 6 \text{ _____ } 4$

(b) $38 + (\text{_____ _____}) = 38 + 9 - 4$

(c) $24 - (6 + 4) = 24 \text{ _____ } 6 - 4$

(d) $24 - 6 - 4 = 24 - 6 \text{ _____ } 4$

(e) $27 - (8 + 3) = 27 \text{ _____ } 8 \text{ _____ } 3$

(f) $27 - (\text{_____ _____}) = 27 - 8 + 3$

Question 2. Remove the brackets and write the expression having the same value.

(a) $14 + (12 + 10)$

(b) $14 - (12 + 10)$

(c) $14 + (12 - 10)$

(d) $14 - (12 - 10)$

(e) $-14 + 12 - 10$

(f) $14 - (-12 - 10)$

Question 3. Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal?

(a) $(6 + 10) - 2$ and $6 + (10 - 2)$

(b) $16 - (8 - 3)$ and $(16 - 8) - 3$

(c) $27 - (18 + 4)$ and $27 + (-18 - 4)$

Question 4. In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms.

(a) $319 + 537, 319 - 537, -537 + 319, 537 - 319$

(b) $87 + 46 - 109, 87 + 46 - 109, 87 + 46 - 109, \ $$ 87 - 46 + 109, \ $$ 87 - (46 + 109), \ $$ (87 - 46) + 109$

Question 5. Add brackets at appropriate places in the expressions such that they lead to the values indicated.

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

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

(c) $-22 - 12 + 10 + 22 = -22$

Question 6. Using only reasoning of how terms change their values, fill the blanks to make the expressions on either side of the equality ($=$) equal.

(a) $423 + \text{______} = 419 + \text{______}$

(b) $207 - 68 = 210 - \text{______}$

Question 7. 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 $3 - (5 - 2) = 0$.

Question 8. 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.

Question 9. 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) $73 - (14 - 1)$

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

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

Question. Lhamo and Norbu went to a hotel. Each of them ordered a vegetable cutlet and a rasgulla. A vegetable cutlet costs $\text{₹}43$ and a rasgulla costs $\text{₹}24$. Write an expression for the amount they will have to pay.

If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid?

Question 1. $5 \times 4 + 3 \neq 5 \times (4 + 3)$. Can you explain why?

Question 2. Is $5 \times (4 + 3) = 5 \times (3 + 4) = (3 + 4) \times 5$?

Question 1. $97 \times 25$ means $97$ times $25$. We can write it as $(100 - 3) \times 25$. We know that this is the same as the difference of $100$ times $25$ and $3$ times $25$:

$97 \times 25 = 100 \times 25 - 3 \times 25$

Find this value.

Question 2. Use this method to find the following products:

(a) $95 \times 8$

(b) $104 \times 15$

(c) $49 \times 50$

Question 3. Is this quicker than the multiplication procedure you use generally? Which other products might be quicker to find like the ones above?

Question 1. 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) $(8 + 3) \times 4 = 8 \times 4 + 3 \times 4$

(c) $3 \times (5 + 8) = 3 \times 5 \text{ _____ } 3 \times \text{____}$

(d) $(9 + 2) \times 4 = 9 \times 4 \text{ _____ } 2 \times\text{____}$

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

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

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

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

(i) $5 \times (9 – 2) = 5 \times 9 – 5 \times \text{____}$

(j) $(5 – 2) \times 7 = 5 \times 7 – 2 \times \text{____}$

(k) $5 \times (8 – 3) = 5 \times 8 \text{ _____ } 5 \times \text{____}$

(l) $(8 – 3) \times 7 = 8 \times 7 \text{ _____ } 3 \times 7$

(m) $5 \times (12 – \text{____}) = \text{____ } 5 \times \text{____}$

(n) $(15 – \text{____}) \times 7 = \text{____ } 6 \times 7$

(o) $5 \times (\text{____} – \text{____}) = 5 \times 9 – 5 \times 4$

(p) $(\text{____} – \text{____}) \times \text{____} = 17 \times 7 – 9 \times 7$

Question 2. 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) $(8 – 3) \times 29$ ________ $(3 – 8) \times 29$

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

(c) $23 \times (17 – 9)$ ________ $23 \times 17 + 23 \times 9$

(d) $(34 – 28) \times 42$ ________ $34 \times 42 – 28 \times 42$

Question 3. Here is one way to make $14$: $2 \times (1 + 6) = 14$. Are there other ways of getting $14$? Fill them out below:

(a) $\text{_____} \times (\text{_____} + \text{_____}) = 14$

(b) $\text{_____} \times (\text{_____} + \text{_____}) = 14$

(c) $\text{_____} \times (\text{_____} + \text{_____}) = 14$

(d) $\text{_____} \times (\text{_____} + \text{_____}) = 14$

Question 4. 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.

Question 1. 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 $9\text{ kg}$ of mangoes each day from his orchard and Shyam supplies $11\text{ kg}$ 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 $\text{₹}20,000$ per month. She spends $\text{₹}5,000$ on rent, $\text{₹}5,000$ on food, and $\text{₹}2,000$ 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 $3\text{ cm}$ up a post, and during the night while asleep, accidentally slips down by $2\text{ cm}$. The post is $10\text{ cm}$ high, and a delicious treat is on its top. In how many days will the snail get the treat?

Question 2. 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) $(7 - 2) \times 8$

(c) $8 \times 7$

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

(e) $7 \times 5 - 2$

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

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

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

Question 3. Find different ways of evaluating the following expressions:

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

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

Question 4. Compare the following pairs of expressions using ‘$<$’, ‘$>$’ or ‘$=$’ or by reasoning.

(a) $49 - 7 + 8$ ________ $49 - 7 + 8$

(b) $83 \times 42 - 18$ ________ $83 \times 40 - 18$

(c) $145 - 17 \times 8$ ________ $145 - 17 \times 6$

(d) $23 \times 48 - 35$ ________ $23 \times (48 - 35)$

(e) $(16 - 11) \times 12$ ________ $-11 \times 12 + 16 \times 12$

(f) $(76 - 53) \times 88$ ________ $88 \times (53 - 76)$

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

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

Question 5. 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) $37 \times 44 + 93 + 76$

Question 6. Choose a number and create ten different expressions having that value.