Ganita Prakash I · Chapter 2 — Arithmetic Expressions

Mallika spends ₹25 on lunch every school day. How much does she spend from Monday to Friday? We can write the answer before working it out: 5 × 25. A mathematical phrase like this, built from numbers and the signs +, −, × and ÷, is called an arithmetic expression.

Value of an expression. Every arithmetic expression stands for one number, called its value. The value of 5 × 25 is 125, and we write 5 × 25 = 125. The sign ‘=’ links an expression to its value. We can also read expressions in words: 13 + 2 is ‘13 plus 2’ or ‘the sum of 13 and 2’; 5 × 25 is ‘5 times 25’ or ‘the product of 5 and 25’.

Many expressions, one value. Different expressions can have the same value. The number 12 can be written as 10 + 2, 15 − 3, 3 × 4 or 24 ÷ 2. Try it with your own favourite number: how many expressions can you make for it?

Comparing expressions. Just as we compare numbers, we can compare expressions by their values, using =, ‘is greater than’ and ‘is less than’. For example, 10 + 2 is greater than 7 + 1, because 12 is greater than 8. And 13 − 2 is less than 4 × 3, because 11 is less than 12.

Filling blanks to balance. When an ‘=’ sign joins two expressions, both sides must have the same value.

• 13 + 4 = ___ + 6. The left side is 17, so the blank is 11.
• 22 + ___ = 6 × 5. The right side is 30, so the blank is 8.
• 8 × ___ = 64 ÷ 2. The right side is 32, so the blank is 4.
• 34 − ___ = 25. The blank is 9.

Putting expressions in order. To arrange 67 − 19, 67 − 20, 35 + 25, 5 × 11 and 120 ÷ 3 from smallest to largest, find each value: 48, 47, 60, 55 and 40. The order is 120 ÷ 3, 67 − 20, 67 − 19, 5 × 11, 35 + 25. Notice that 67 − 20 is less than 67 − 19 without any working: taking away more leaves less.

Why expressions are useful. An expression records how a quantity is made, not just the final number. ‘5 × 25’ tells us five days at ₹25 each. If the price changes, we can see at once what to change. Throughout this chapter we will read expressions carefully, find their values and learn tools that make them clear.

Which is greater: 1023 + 125 or 1022 + 128? You could add both, but there is a smarter way. Imagine a story.

A marble story. Raja had 1023 marbles and got 125 more. Joy had 1022 marbles and got 128 more. At the start Raja had 1 marble more than Joy. But today Joy got 3 more than Raja. So Joy ends with 3 − 1 = 2 more marbles. Therefore 1023 + 125 is less than 1022 + 128.

A subtraction story. Which is greater: 113 − 25 or 112 − 24? Raja had 113 marbles and lost 25; Joy had 112 and lost 24. Raja started with 1 more, but he also lost 1 more. So they end up with the same number: 113 − 25 = 112 − 24 (both are 88).

The key idea. Compare each part of one expression with the matching part of the other, and think about how each change affects the value.

• In a sum, if one number goes up by some amount and the other goes down by the same amount, the sum does not change.
• In a difference, if both numbers go up (or both go down) by the same amount, the difference does not change.
• In a difference, taking away a bigger number leaves less.

Practice with reasoning.

Check one. For the last row, 213 − 77 = 136 and 214 − 76 = 138. The second is 2 more, just as the reasoning said: 1 more at the start and 1 less taken away.

Why bother? Reasoning is often faster than calculating, and it shows why one value is bigger. It also builds the habit of looking at the structure of an expression, which we will need when the numbers become too large to work out quickly, or when letters take the place of numbers in later chapters.

In language, a comma can change meaning. ‘Shalini sat next to a friend with toys’ suggests the friend has the toys. ‘Shalini sat next to a friend, with toys’ suggests Shalini brought them. Expressions can be unclear in the same way.

A confusing expression. Mallesh brought 30 marbles. Arun brought 5 bags with 4 marbles in each. Mallesh wrote the total as 30 + 5 × 4. A friend added first: 30 + 5 = 35, then 35 × 4 = 140. Mallesh multiplied first: 5 × 4 = 20, then 30 + 20 = 50. The story tells us that 50 is right, but the expression by itself does not say which operation comes first.

Brackets remove the doubt. We write 30 + (5 × 4). The rule is: work out what is inside the brackets first, then do the rest. So 30 + (5 × 4) = 30 + 20 = 50.

When brackets really matter. Irfan buys biscuits for ₹15 and toor dal for ₹56 and pays with a ₹100 note. Is his change 100 − 15 + 56? Working left to right gives 85 + 56 = 141, more money than he paid! The total cost 15 + 56 must be taken away as one amount, so we write 100 − (15 + 56) = 100 − 71 = 29. Irfan gets ₹29 back.

Writing brackets from a story. Ask: which amount must be found as a whole before it is used? Put that amount in brackets.

• Two friends each buy a ₹43 cutlet and a ₹24 sweet: each share is 43 + 24, so the total is 2 × (43 + 24).
• A drone rises 6 m then falls 4 m, five times: 5 × (6 − 4) = 10 m higher.
• From 200 pages, a student reads 40 pages and then 3 more: 200 − (40 + 3) pages are left.

Same numbers, different values. 30 + (5 × 4) = 50 but (30 + 5) × 4 = 140. 100 − (15 + 56) = 29 but (100 − 15) + 56 = 141. Moving the brackets changes the meaning, so always check that your brackets match the situation.

Nested brackets. Sometimes one bracket sits inside another, like 500 − (250 − (60 + 40)). Work from the innermost bracket outwards: 60 + 40 = 100, then 250 − 100 = 150, then 500 − 150 = 350.

Brackets are one tool for clearing up order. In the next topic we meet a second tool, terms, which tells us what to do when there are no brackets.

What if an expression has no brackets, like 30 + 5 × 4? Mathematicians agreed on a second tool: terms.

What is a term? Terms are the parts of an expression that are joined by ‘+’ signs. In 12 + 7, the terms are 12 and 7.

Subtraction as adding the opposite. Taking away 14 is the same as adding −14 (the inverse of 14). So 83 − 14 = 83 + (−14), and its terms are 83 and −14. We rewrite every subtraction this way to find the terms:

A product like 6 × 3 or 4 × 6 has no ‘+’ inside it, so it is a single term.

The rule for evaluating. First find the value of each term, then add the terms. So 30 + 5 × 4 has terms 30 and 5 × 4. The second term is 20, and 30 + 20 = 50, which matches the marble story. Brackets inside a term are worked out first: in 5 × (3 + 2) + 7 × 8 + 3 the terms are 5 × (3 + 2) = 25, 7 × 8 = 56 and 3, so the value is 25 + 56 + 3 = 84.

Terms from stories.

• Four friends order dosas at ₹23 each and leave a ₹5 tip: 4 × 23 + 5. Terms: 4 × 23 and 5. Value: 92 + 5 = 97 rupees.
• 33 children form groups of 5 in a game: 6 groups and 3 children left over, written 6 × 5 + 3. With groups of 4 it is 8 × 4 + 1; with groups of 7 it is 4 × 7 + 5.
• A trader has 4 rice packets and packs 100 kg more into 2 kg packets: 4 + 100 ÷ 2 = 4 + 50 = 54 packets.
• ₹432 paid as 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1 has four terms, one for each kind of note or coin.

More practice. 39 − 2 × 6 + 11 has terms 39, −2 × 6 and 11, so the value is 39 − 12 + 11 = 38. 48 − 10 × 2 + 16 ÷ 2 has terms 48, −10 × 2 and 16 ÷ 2, giving 48 − 20 + 8 = 36. 6 × 3 − 4 × 8 × 5 gives 18 − 160 = −142.

Why terms work. A term is something already ‘packed together’, like a number of groups of the same size. We must count each pack before we combine the packs. That is why × and ÷ inside a term are done before the + between terms.

Once an expression is written as a sum of terms, does the order of adding matter? Let us test it, including negative terms.

Swapping two terms. A drone goes 6 m up and then 4 m down: 6 + (−4) = 2. Swap the terms: (−4) + 6 = 2. Same value. Try others: (−4) + (−2) = −6 and (−2) + (−4) = −6. Swapping two terms never changes the sum:

Why it works. Think of positive and negative tokens. 6 + (−4) is 6 positive tokens and 4 negative tokens. Each positive cancels a negative, leaving 2 positives. It does not matter which pile you put down first.

Grouping three terms. Take (−7) + 10 + (−11). Add the first two first: 3 + (−11) = −8. Or add the last two first: (−7) + (−1) = −8. Or even add the two negatives first: (−18) + 10 = −8. Every way gives −8.

Swapping is called the commutative property of addition, and grouping is the associative property of addition. Together they say: the terms of a sum can be added in any order.

Using it. Manasa added a long list and got 11,749, then saw she had missed the number 9055. She does not need to start again. She simply adds 9055 to her total: 11,749 + 9055 = 20,804. Grouping allows her to add the missing term last.

Smart adding. Look for terms that pair up nicely: 37 + 58 + 63 = (37 + 63) + 58 = 100 + 58 = 158. In 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10, pair each positive with the negative after it: five pairs of −1, giving −5.

Order in everyday life. Some jobs can be done in any order. Putting on a hat and shoes gives the same result either way. Others cannot: socks must go on before shoes. Addition of terms is like the hat and shoes. Subtraction is like the socks and shoes: 10 − 3 is not 3 − 10. That is exactly why we rewrite subtraction as adding a negative term before we swap anything: 10 − 3 = 10 + (−3) = (−3) + 10.

Sometimes it is easier to get rid of brackets and work step by step. But we must be careful about the signs.

A minus sign before brackets. Find 200 − (40 + 3). Working the bracket gives 200 − 43 = 157. Another way: take away 40 first (160), then take away 3 more (157). So 200 − (40 + 3) = 200 − 40 − 3. Both numbers in the bracket are taken away. It is not 200 − 40 + 3, which gives 163.

Irfan's change works the same way: 100 − (15 + 56) = 100 − 15 − 56 = 85 − 56 = 29. First the shopkeeper owes him 85 for the biscuits alone; then the dal's ₹56 comes out of that.

A minus inside the bracket too. Find 500 − (250 − 100). The bracket is 150, so the value is 350. If we take away 250 directly, we have taken away 100 too many, so we must add 100 back: 500 − 250 + 100 = 350. Thus 500 − (250 − 100) = 500 − 250 + 100. (Check that 500 − 250 − 100 = 150 is wrong.)

The pattern. When brackets with a minus sign in front are removed, every term inside changes its sign: + becomes − and − becomes +.

A plus sign before brackets. Hira has 28 coins in one bag and 35 in another, and gives away 10 from the second: 28 + (35 − 10). The bracket is 35 + (−10), and terms can be added in any order, so this is 28 + 35 − 10 = 53. When the brackets have a plus sign in front, the terms keep their signs.

Do not just memorise. Instead of learning the rule by heart, think about what the expression means. ‘Take away 12 − 10’ means taking away 2. Taking away 12 removes 10 too many, so add 10 back. The meaning gives the rule.

Putting brackets in. The idea also works backwards. To make 34 − 9 + 12 equal 13, put brackets so that 9 and 12 are both taken away: 34 − (9 + 12) = 13. To make 56 − 14 − 8 equal 50, write 56 − (14 − 8) = 56 − 6 = 50.

If we know the value of an expression, we can often find the value of a slightly different one without starting again. We just ask how one term has changed.

Changing one term in a sum. We know 53 + (−16) = 37.

• 54 + (−16): 54 is one more than 53, so the value is one more: 38.
• 52 + (−16): one less, so 36.
• 53 + (−15): −15 is one more than −16 (it is to the right on the number line), so the value is 38.
• 53 + (−17): −17 is one less than −16, so the value is 36.

Changing two terms. If one term goes up by 1 and another goes down by 1, the sum is unchanged. −87 + (−16) = −103, and −88 + (−15) is also −103: −88 is one less than −87, and −15 is one more than −16.

A shortcut for subtracting 9. Jasoda subtracts 9 by taking away 10 and adding 1: 36 − 9 = 26 + 1 = 27. Is she always right? Yes, because 9 = 10 − 1, so n − 9 = n − (10 − 1) = n − 10 + 1. In the same way, n − 99 = n − 100 + 1 and n + 98 = n + 100 − 2.

Keeping a difference the same. 207 − 68 = 210 − ___? The first number went up by 3. For the difference to stay the same, the number taken away must also go up by 3: 68 + 3 = 71. So 207 − 68 = 210 − 71 (both are 139).

Keeping a sum the same. 423 + ___ = 419 + ___. The first number on the right is 4 less, so its partner must be 4 more. Any pair like 423 + 10 = 419 + 14 works.

Why this is useful. Tinkering helps with mental arithmetic (subtract 10 and add 1 is easier than subtracting 9), and it helps us check answers. If you know one value, you can predict the next one. It is also good practice in thinking about expressions as made of terms, each with its own effect.

Lhamo and Norbu each order a cutlet (₹43) and a rasgulla (₹24). Is the bill 2 × 43 + 24? No: that means ‘24 more than twice 43’, which is only ₹110. Each person's share is 43 + 24, and we want twice that: 2 × (43 + 24) = 2 × 67 = ₹134.

Another way to see the same bill. Together they bought two cutlets and two rasgullas: 2 × 43 + 2 × 24 = 86 + 48 = ₹134. So 2 × (43 + 24) = 2 × 43 + 2 × 24.

Rows in a parade. 4 rows of 5 scouts and 3 rows of 5 guides march together. Count the groups separately: 4 × 5 + 3 × 5 = 20 + 15 = 35. Or count all the rows first: (4 + 3) × 5 = 7 × 5 = 35. Same answer.

The general idea. 10 × 98 + 3 × 98 means ten 98s and then three more 98s: that is 13 × 98. In the same way, 14 × 10 − 6 × 10 is 8 × 10. In short: a multiple of a sum (or difference) equals the sum (or difference) of the multiples. This is the distributive property.

Quick multiplication.

• Given 53 × 18 = 954, find 63 × 18: (53 + 10) × 18 = 954 + 180 = 1134.
• 97 × 25 = (100 − 3) × 25 = 2500 − 75 = 2425.
• 95 × 8 = 800 − 40 = 760; 104 × 15 = 1500 + 60 = 1560; 49 × 50 = 2500 − 50 = 2450.

Word problems. Two farmers send 9 kg and 11 kg of mangoes every day for a week: (9 + 11) × 7 = 140 kg. Binu earns ₹20,000 a month and spends ₹5000, ₹5000 and ₹2000: she saves (20,000 − 5000 − 5000 − 2000) × 12 = ₹96,000 a year. A boy reads one story a day except on two days a week: in 8 weeks he reads (7 − 2) × 8 = 7 × 8 − 2 × 8 = 40 stories.

Watch out. 5 × 4 + 3 is 23, but 5 × (4 + 3) is 35. Without brackets the multiplier reaches only one term. Also, 23 × (17 − 9) equals 23 × 17 − 23 × 9, not 23 × 17 + 23 × 9.