Class 7 Maths Chapter 2: Arithmetic Expressions Notes

What is this chapter about?

You have been writing things like 13 + 2 and 12 × 5 for years. This chapter looks closely at such arithmetic expressions: what their values are, how to compare them, and how to read a long expression so that everyone gets the same answer. Along the way it names three properties you have been using without names: the commutative, associative and distributive properties.

Simple expressions and their values

Every expression has a value, the number it works out to, and the equals sign joins the two: 13 + 2 = 15. Mallika pays ₹25 for lunch on each of 5 school days, so her week’s lunch costs 5 × 25. Many expressions share one value: 10 + 2, 15 − 3, 3 × 4 and 24 ÷ 2 are all 12.

Comparing expressions by reasoning

Expressions can be compared with =, < and >, often without working them out. Raja has 1023 marbles and gets 125 more; Joy has 1022 and gets 128 more. Raja starts 1 ahead, but Joy gains 3 more, so 1023 + 125 < 1022 + 128. And 113 − 25 = 112 − 24, because the one who starts 1 ahead also loses 1 more. Read more in reading and evaluating expressions.

Reading complex expressions

A sentence can mean two things until a comma settles it. Expressions are similar. Mallesh brought 30 marbles, and Arun brought 5 bags of 4, so Mallesh wrote 30 + 5 × 4. Purna added first and got 140; Mallesh multiplied first and got 50, which is right. Mathematics uses two tools to remove such confusion:

• Brackets: work out what is inside them first. Irfan pays ₹100 for biscuits (₹15) and toor dal (₹56), so his change is 100 − (15 + 56) = 29 rupees.
• Terms: the parts of an expression joined by + signs. Each subtraction is written as adding the inverse, so 83 − 14 has the terms 83 and −14, and 6 × 5 + 3 has the terms 6 × 5 and 3.

To find a value, work out the brackets, then each term, and add the terms.

Swapping and grouping terms

A drone goes 6 m up and 4 m down: 6 − 4 = 6 + (−4) = 2, and swapping the terms, (−4) + 6, also gives 2. With three terms, (−7) + 10 + (−11) is −8 whichever two terms you add first. So terms can be added in any order: swapping is the commutative property of addition and grouping is the associative property. Everyday actions are not always like this: hat and shoes can go on in either order, but socks must go on before shoes. See terms and brackets.

Removing brackets

200 − (40 + 3) can be worked out as 200 − 40 − 3: both numbers are taken away. In 500 − (250 − 100), taking away all of 250 removes 100 too many, so 100 is added back: 500 − 250 + 100. So a minus sign before brackets flips the sign of every term inside, while a plus sign leaves them as they are: Hira’s coins, 28 + (35 − 10), are 28 + 35 − 10 = 53.

Tinker the terms

Change one term and the value changes by the same amount. Since 53 + (−16) = 37, the expression 54 + (−16) is 38, and so is 53 + (−15), because −15 is 1 more than −16. Jasoda uses the same idea to subtract 9: she subtracts 10 and adds 1, because 36 − 9 = 36 − (10 − 1) = 36 − 10 + 1.

The distributive property

Lhamo and Norbu each order a cutlet (₹43) and a rasgulla (₹24). Together they pay 2 × (43 + 24), which equals 2 × 43 + 2 × 24 = 134 rupees. Writing 2 × 43 + 24 would pay for only one rasgulla. In a parade, 4 rows of 5 scouts and 3 rows of 5 guides make 4 × 5 + 3 × 5 = (4 + 3) × 5 = 35 children. The general rule, the multiple of a sum (or difference) is the sum (or difference) of the multiples, is called the distributive property. It also makes products quick: since 53 × 18 = 954, 63 × 18 = 954 + 10 × 18 = 1134, and 97 × 25 = 100 × 25 − 3 × 25 = 2425. Read the distributive property.

Expression Engineer

The chapter ends with puzzles: make expressions from three 3s, from four 4s, from 1, 2, 3, 4 and 5, and from all the digits 0 to 9. For example, 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 = 100. Answers to every question, the Figure it Out exercises and the puzzles are in the NCERT solutions for Chapter 2.