Operations with Multiple Rational Numbers
Operations with multiple rational numbers is a Grade 7 skill in Big Ideas Math, Course 2 involving chains of addition, subtraction, multiplication, and division applied to fractions, decimals, and integers. The order of operations (PEMDAS) governs the sequence: parentheses first, then multiplication and division left to right, then addition and subtraction left to right. Sign rules apply: the product of two negatives is positive, and the product of one negative and one positive is negative. For example, (−2/3)(−3/4) + 1/2 = 1/2 + 1/2 = 1. Keeping track of signs at each step and simplifying fractions before combining reduces computational errors.
Key Concepts
Property To evaluate expressions with multiple rational numbers, first convert all numbers (decimals, integers) to fractions. Then, perform the multiplications and divisions from left to right, remembering that dividing by a number is the same as multiplying by its reciprocal. The sign of the final answer is positive if there is an even number of negative factors and negative if there is an odd number.
Examples $ \frac{3}{4} \cdot 1.2 \div ( 3) = \frac{3}{4} \cdot \frac{12}{10} \div ( \frac{3}{1}) = \frac{3}{4} \cdot \frac{6}{5} \cdot ( \frac{1}{3}) = \frac{18}{60} = \frac{3}{10}$ or $0.3$ $5 \cdot ( \frac{2}{5}) \div 0.25 = \frac{5}{1} \cdot ( \frac{2}{5}) \cdot \frac{1}{0.25} = \frac{5}{1} \cdot ( \frac{2}{5}) \cdot \frac{1}{\frac{1}{4}} = \frac{5}{1} \cdot ( \frac{2}{5}) \cdot 4 = 8$.
Explanation When multiplying and dividing several rational numbers, it is often easiest to convert all numbers into fractions first. Determine the sign of the final result by counting the number of negative terms. Then, change all division operations to multiplication by the reciprocal of the following number. Finally, multiply the numerators and denominators, simplifying the result if possible.
Common Questions
What governs the sequence of operations with multiple rational numbers?
The order of operations (PEMDAS): Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right).
What is the sign rule for multiplying rational numbers?
Negative × Negative = Positive. Positive × Negative = Negative. Apply this rule at each multiplication or division step.
How do you evaluate (−2/3)(−3/4) + 1/2?
Multiply first: (−2/3)(−3/4) = 6/12 = 1/2 (positive, since both negative). Then add: 1/2 + 1/2 = 1.
When should you simplify fractions during a multi-step calculation?
Simplify (cancel common factors) before multiplying to keep numbers small and reduce errors. For addition/subtraction, simplify after finding the common denominator.
How do you handle a chain like −1.5 + 0.5 × (−2)?
Multiply first (order of operations): 0.5 × (−2) = −1. Then add: −1.5 + (−1) = −2.5.
What is a common mistake with multiple operations and rational numbers?
Adding before multiplying (ignoring PEMDAS), or losing track of signs when multiplying two negatives and incorrectly making the result negative.