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Mathematics: Class 7 (Part 1) — Chapter 2: ARITHMETIC EXPRESSIONS

2.1 Simple Expressions

Mathematical phrases like $13 + 2$, $20 - 4$, $12 \times 5$, and $18 \div 3$ are called arithmetic expressions.

Example 1: Mallika's Lunch Spending

Mallika spends ₹25 every day for lunch at school. For a week (Monday to Friday), the expression for the total amount spent is $5 \times 25$.

  • This can be read as "5 times 25" or "the product of 5 and 25".

Different expressions can have the same value. For example, the number 12 can be expressed as $10 + 2$, $15 - 3$, $3 \times 4$, or $24 \div 2$.

Comparing Expressions

Expressions can be compared based on their values using '$=$', '$<$', and '$>$' signs.

  • Example: $10 + 2 > 7 + 1$ because $12 > 8$.
  • Example: $13 - 2 < 4 \times 3$ because $11 < 12$.

Figure it Out (Activities)

  • Fill in the blanks to make expressions equal on both sides of the '$=$' sign.
  • Arrange given expressions in ascending (increasing) order of their values.

Example 2: Comparing $1023 + 125$ and $1022 + 128$

By imagining marbles, if Raja has 1 more marble initially but Joy gets 3 more than Raja, Joy ends up with 2 more marbles. Thus, $1023 + 125 < 1022 + 128$.

Example 3: Comparing $113 - 25$ and $112 - 24$

By imagining marbles, if Raja has 1 more marble initially but loses 1 more than Joy, they end up with an equal number of marbles. Thus, $113 - 25 = 112 - 24$.

2.2 Reading and Evaluating Complex Expressions

When an expression has multiple operations and no specified order, there can be confusion in evaluating its value. Just as punctuation resolves ambiguity in language, brackets and the notion of terms are used in mathematics.

Example 4: Mallesh and Arun's Marbles

Mallesh wrote $30 + 5 \times 4$. This can be interpreted in two ways:

  • $(30 + 5) \times 4 = 35 \times 4 = 140$ (Purna's interpretation)
  • $30 + (5 \times 4) = 30 + 20 = 50$ (Mallesh's correct interpretation, as Arun brought 5 bags of 4 marbles each).

Brackets in Expressions

Brackets clarify the order of operations. When evaluating an expression with brackets, first find the value of the expressions inside the brackets.

  • Example: $30 + (5 \times 4) = 30 + 20 = 50$.

Example 5: Irfan's Change

Irfan bought biscuits for ₹15 and toor dal for ₹56, paying ₹100. The change is $100 - (15 + 56)$.

  • Evaluating inside brackets first: $100 - 71 = 29$.
  • If brackets are ignored: $100 - 15 + 56 = 85 + 56 = 141$ (absurd).

Rule: Upon removing brackets preceded by a negative sign, the signs of the terms inside the brackets change.

  • $100 - (15 + 56) = 100 - 15 - 56$.

Terms in Expressions

Terms are the parts of an expression separated by a '$+$' sign.

  • Example: In $12 + 7$, the terms are $12$ and $7$.
  • Subtracting a number is the same as adding its inverse. For example, $83 - 14 = 83 + (-14)$. The terms are $83$ and $-14$.
  • All subtractions are converted to additions to identify terms.
  • Example: $-18 - 3 = -18 + (-3)$.
  • Example: $6 \times 5 + 3$. Terms are $6 \times 5$ and $3$. Note that $6 \times 5$ is a single term.
  • Example: $2 - 10 + 4 \times 6 = 2 + (-10) + (4 \times 6)$. Terms are $2$, $-10$, and $4 \times 6$.

Swapping and Grouping Terms

For expressions with only additions (after converting subtractions), changing the order or grouping of terms does not change the value of the expression.

  • Swapping: For two terms, $Term 1 + Term 2 = Term 2 + Term 1$.
  • Grouping: For three terms, $(Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3)$.
  • This is due to the commutative property of addition and the associative property of addition, respectively.

When expressions involve multiplication and division without specified order, evaluate the terms first, then add them.

  • Example: $30 + 5 \times 4 = 30 + (5 \times 4) = 30 + 20 = 50$.
  • Example: $5 \times (3 + 2) + 7 \times 8 + 3 = (5 \times 5) + (7 \times 8) + 3 = 25 + 56 + 3 = 84$.

More Expressions and Their Terms

Example 7: Hotel Bill

Amu, Charan, Madhu, and John ordered four dosas at ₹23 each and tipped ₹5. The total cost is $4 \times 23 + 5 = 92 + 5 = 97$. The terms are $4 \times 23$ and $5$.

Example 9: Raghu's Rice Packets

Raghu had 4 packets and bought 100 kg of rice, packed into 2 kg packets. The number of new packets is $100 \div 2$. Total packets: $4 + \frac{100}{2}$. The terms are $4$ and $\frac{100}{2}$.

Example 10: Kannan's Payment

To pay ₹432, Kannan can use various combinations of notes and coins. For example:

  • $4 \times 100 + 1 \times 20 + 1 \times 10 + 2 \times 1$. Terms: $4 \times 100$, $1 \times 20$, $1 \times 10$, $2 \times 1$.
  • $8 \times 50 + 1 \times 10 + 4 \times 5 + 2 \times 1$. Terms: $8 \times 50$, $1 \times 10$, $4 \times 5$, $2 \times 1$.

Removing Brackets—I (Negative Sign Precedence)

When removing brackets preceded by a negative sign, the signs of the terms inside the brackets change.

  • Example: $200 - (40 + 3) = 200 - 40 - 3 = 157$.
  • Example: $500 - (250 - 100) = 500 - 250 + 100 = 350$.

When brackets are NOT preceded by a negative sign, the terms within them do not change their signs.

  • Example: $28 + (35 - 10) = 28 + 35 - 10 = 53$.

Removing Brackets—II (Distributive Property)

The distributive property states that the multiple of a sum (or difference) is the same as the sum (or difference) of the multiples.

  • Example: $2 \times (43 + 24) = 2 \times 43 + 2 \times 24$.
  • Example: $4 \times 5 + 3 \times 5 = (4 + 3) \times 5$.
  • Example: $10 \times 98 + 3 \times 98 = (10 + 3) \times 98$.
  • Example: $14 \times 10 - 6 \times 10 = (14 - 6) \times 10$.

Example 17: Using Distributive Property for Multiplication

Given $53 \times 18 = 954$. To find $63 \times 18$:

  • $63 \times 18 = (53 + 10) \times 18 = 53 \times 18 + 10 \times 18 = 954 + 180 = 1134$.

Example 18: Evaluating $97 \times 25$

To find $97 \times 25$:

  • $97 \times 25 = (100 - 3) \times 25 = 100 \times 25 - 3 \times 25 = 2500 - 75 = 2425$.

This method is quicker when one of the numbers is close to a multiple of 10, 50, 100, etc.

Figure it Out (Activities)

  • Write expressions for given situations and find their values.
  • Find the sum of numbers in pictures in at least two different ways.
  • Identify expressions that describe a given scenario.
  • Find different ways of evaluating expressions.
  • Compare pairs of expressions using reasoning.
  • Identify expressions with the same value without computation.
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