The Number System

Operations with Rational Numbers

Grade 7

✓By the end of this lesson students will be able to add rational numbers, including positive and negative fractions and decimals.
✓By the end of this lesson students will be able to subtract rational numbers, including positive and negative fractions and decimals.
✓By the end of this lesson students will be able to multiply rational numbers, including positive and negative fractions and decimals.
✓By the end of this lesson students will be able to divide rational numbers, including positive and negative fractions and decimals.
✓By the end of this lesson students will be able to solve real-world and mathematical problems involving the four operations with rational numbers.

Key concepts

Rational Number

A rational number is any number that can be written as a fraction p/q where p and q are integers and q is not zero. This includes integers, fractions, terminating decimals, and repeating decimals.

Adding Rational Numbers

To add rational numbers:\n\n* **Same Signs**: Add the absolute values of the numbers and keep the common sign.\n* **Different Signs**: Subtract the smaller absolute value from the larger absolute value. The sign of the answer is the same as the sign of the number with the larger absolute value.\n\nWhen adding fractions, find a common denominator first.

Subtracting Rational Numbers

To subtract a rational number, add its opposite. This means changing the subtraction sign to an addition sign and changing the sign of the number being subtracted. Then, follow the rules for adding rational numbers.

a - b = a + (-b)

Multiplying Rational Numbers

To multiply rational numbers:\n\n* **Same Signs**: The product of two rational numbers with the same sign (both positive or both negative) is positive.\n* **Different Signs**: The product of two rational numbers with different signs (one positive and one negative) is negative.\n\nWhen multiplying fractions, multiply the numerators and multiply the denominators.

Dividing Rational Numbers

To divide rational numbers:\n\n* **Same Signs**: The quotient of two rational numbers with the same sign is positive.\n* **Different Signs**: The quotient of two rational numbers with different signs is negative.\n\nTo divide by a rational number, multiply by its reciprocal. The reciprocal of a fraction a/b is b/a.

a ÷ b = a × (1/b) (where b ≠ 0)

Key facts to remember

1A rational number can be written as a fraction p/q where p and q are integers and q ≠ 0.
2To add numbers with the same sign, add their absolute values and keep the common sign.
3To add numbers with different signs, subtract their absolute values and use the sign of the number with the greater absolute value.
4To subtract a rational number, add its opposite: a - b = a + (-b).
5The product or quotient of two rational numbers with the same sign is positive.
6The product or quotient of two rational numbers with different signs is negative.
7To divide by a rational number, multiply by its reciprocal.
8When performing addition or subtraction with fractions, always find a common denominator first.

Worked examples

Example 1

Calculate: -3.75 + 1/2 - (-2.25)

IConvert the fraction to a decimal to work with consistent forms: 1/2 = 0.5.

IIRewrite the expression: -3.75 + 0.5 - (-2.25).

IIIAddress the subtraction of a negative number (add the opposite): -3.75 + 0.5 + 2.25.

IVPerform addition from left to right: -3.75 + 0.5. Since signs are different, subtract absolute values (3.75 - 0.5 = 3.25) and keep the sign of the number with the larger absolute value (-3.75), resulting in -3.25.

VContinue with the remaining addition: -3.25 + 2.25. Since signs are different, subtract absolute values (3.25 - 2.25 = 1.00) and keep the sign of the number with the larger absolute value (-3.25), resulting in -1.00.

Answer

-1

It's often helpful to convert all numbers to the same form (all decimals or all fractions) before performing operations.

Example 2

Evaluate: (-2/3) ÷ (4/5) × (-1/2)

IPerform division first by multiplying by the reciprocal of the divisor: (-2/3) × (5/4) × (-1/2).

IIMultiply the first two fractions: (-2/3) × (5/4). Multiply numerators (-2 × 5 = -10) and denominators (3 × 4 = 12). This gives -10/12.

IIISimplify the resulting fraction: -10/12 simplifies to -5/6.

IVNow multiply the simplified result by the last fraction: (-5/6) × (-1/2). Multiply numerators (-5 × -1 = 5) and denominators (6 × 2 = 12).

VThe product of two negative numbers is positive.

Answer

5/12

Remember to simplify fractions before or after multiplication to make calculations easier.

Example 3

A submarine is at -25.5 feet relative to sea level. It then descends 15.2 feet, and later ascends 10.8 feet. What is its final depth?

IIdentify the initial position: -25.5 feet.

II"Descends 15.2 feet" means the depth decreases, which can be represented as adding -15.2 feet: -25.5 + (-15.2).

III"Ascends 10.8 feet" means the depth increases, which can be represented as adding 10.8 feet: + 10.8.

IVCombine the operations into a single expression: -25.5 + (-15.2) + 10.8.

VAdd the first two numbers: -25.5 + (-15.2). Since signs are the same, add absolute values (25.5 + 15.2 = 40.7) and keep the common sign, resulting in -40.7.

VIAdd the last number: -40.7 + 10.8. Since signs are different, subtract absolute values (40.7 - 10.8 = 29.9) and keep the sign of the number with the larger absolute value (-40.7), resulting in -29.9.

Answer

The final depth of the submarine is -29.9 feet.

Representing 'descends' as adding a negative number helps maintain consistency with addition rules for signed numbers.

Common mistakes

✗**Sign Errors**: Incorrectly applying the rules for signs, especially with subtraction of negative numbers or multiplication/division.
✗**Forgetting Common Denominators**: Adding or subtracting fractions without first finding a common denominator.
✗**Incorrect Reciprocal**: When dividing fractions, forgetting to multiply by the reciprocal of the *second* fraction, or incorrectly flipping the first fraction instead.
✗**Order of Operations**: Not following the correct order of operations (PEMDAS/BODMAS) when multiple operations are present.
✗**Decimal vs. Fraction Conversion**: Making errors when converting between decimals and fractions, or mixing forms within a calculation.

Exam tips

★**Show All Work**: Write down each step clearly, especially when dealing with multiple operations or negative numbers. This helps catch errors and allows for partial credit.
★**Check Your Signs**: Double-check the sign of your answer after each operation, particularly with multiplication and division of negative numbers.
★**Convert Consistently**: Decide whether to work with decimals or fractions for a problem and convert all numbers to that form before starting calculations.
★**Simplify Fractions**: Simplify fractions before or after operations to make calculations easier and ensure the answer is in simplest form.

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