Revisiting Arithmetic and Like Terms

4.2 Revisiting Arithmetic Expressions

We learnt to work with expressions made of numbers and applied two methods: swapping (adding two numbers in any order) and grouping (adding two numbers grouped with brackets). When performing arithmetic, brackets govern the order in which operations are carried out, and the distributive property^? tells us multiplication distributes over addition.

Let us revise these ideas by finding the easy way of evaluating:

1. \(23 - 10 + 2\)
2. \(83 - 28 - 13 + 32\)
3. \(34 - 14 + 20\)
4. \(42 + 15 - (8 - 7)\)
5. \(68 - (18 + 13)\)
6. \(7 \times 4 + 9 \times 6\)
7. \(20 \times 8 + (16 - 4)\)

Let us evaluate \(23 - 10 + 2\). In terms of signs the number \(23\) is positive, \(10\) is subtracted, and \(2\) is added before we combine two terms. The result is \(23 - 10 + 2 = 23 - 8 = 15\) — which equals \(23 + 2 - 10 = 15\). So terms keep their signs and can be rearranged.

Example 5 — Using Terms to Simplify

Consider \(34 - 14 + 20\). Terms: \(+34, -14, +20\). Rearrange: \(+34 + 20 - 14 = 54 - 14 = 40\).

Consider \(42 + 15 - (8 - 7)\). Inside brackets first: \(8 - 7 = 1\). Then \(42 + 15 - 1 = 56\).

4.3 Omitting the Multiplication Sign

When we multiply a number by a letter-number, we drop the ×:

• \(3 \times a\) is written \(3a\).
• \(a \times b\) is written \(ab\).
• \(5 \times x \times y\) is written \(5xy\).

But note: \(3 \times 4\) is not written \(34\) — otherwise it looks like the two-digit number thirty-four. We never drop × between two numbers, only between a number and letter, or between two letters.

4.4 Like Terms and Unlike Terms

Consider the expression for the area of rectangle AEFD in Fig 4.4. It can be found in two ways: (i) using the side-lengths 8 and \((12-4)\), or (ii) subtracting the area of rectangle EBCF from ABCD.

Method 1: area AEFD = \(8 \times (12-4) = 8n\) with \(n=8\), so area = 64.
Method 2: area AEFD = \(8 \times 12 - 8 \times 4 = 96 - 32 = 64\). Both match.

In general, \(8n\) and \((8 \times 12 - 8 \times 4) = 8(12-4)\) give the same value. Terms such as \(8n, 3n, 12n\) that look like a number multiplying the same letter-number are called like terms^?. They can be combined together by adding or subtracting their coefficients.

Terms with different letter-numbers such as \(8n\) and \(5m\), or \(3x\) and \(3y\), are called unlike terms and cannot be combined.

Example 7 — Furniture Shop

A shop rents out chairs and tables for a day's use. To rent a chair costs ₹40 and to rent a table costs ₹75. The rented furniture must later be returned. For rented items, the shopkeeper pays back some amount per piece: ₹6 per chair and ₹10 per table.

If \(x\) chairs and \(y\) tables were rented at the beginning of the day, and the same amount was returned at the end, the total amount paid to the shopkeeper (net) can be worked out like this:

Total = \((40x + 75y) - (6x + 10y) = 40x - 6x + 75y - 10y = 34x + 65y\).

Here \(40x\) and \(6x\) are like terms (both \(x\)), \(75y\) and \(10y\) are like terms. Unlike terms \(34x\) and \(65y\) cannot be combined; the simplified form is \(34x + 65y\).

Example — Matchstick Triangle Patterns

Consider a pattern where step 1 has 1 triangle (3 sticks), step 2 has 2 triangles (5 sticks), step 3 has 3 triangles (7 sticks), and so on.

At each step, 2 matchsticks are added. For step \(n\), the count is \(3 + 2(n-1) = 2n + 1\). Step 33 has \(2 \times 33 + 1 = 67\) matchsticks.

Figure it Out