Four Operations with Positive & Negative Fractions and Decimals - Seventh Grade Mathematics
The universal sign rules of integer arithmetic apply directly to fractions and decimals, enabling comprehensive computation across the full domain of rational numbers.
Adding and Subtracting Signed Fractions
Adding and subtracting signed fractions requires combining fraction arithmetic (finding common denominators) with the sign rules for integers.
ADDING SIGNED FRACTIONS PROTOCOL
1. Express any mixed numbers as improper fractions.
2. Find the least common denominator (LCD) and rewrite fractions.
3. Apply the integer sign rules to the numerators over the common denominator.
4. Simplify the resulting fraction to lowest terms.
Consider the problem: negative two-thirds plus positive three-fourths. First, determine the least common denominator of three and four, which is twelve. Rewrite negative two-thirds as negative eight-twelfths, and rewrite positive three-fourths as positive nine-twelfths. Now combine the numerators: negative eight plus nine equals positive one. The common denominator remains twelve, resulting in positive one-twelfth.
Example 1: (-2/3) + (3/4) = (-8/12) + (9/12) = (-8 + 9) / 12 = 1/12
Example 2: (-5/6) - (1/4) = (-5/6) + (-1/4) = (-10/12) + (-3/12) = -13/12
When subtracting a fraction, always rewrite the operation as adding the opposite before combining numerators.
Adding and Subtracting Signed Decimals
When computing with signed decimals, align the decimal points vertically to ensure corresponding place values match, then apply the integer rules based on the signs of the numbers.
Example: Evaluate (-14.35) + 8.6
1. The signs are different; subtract smaller absolute value from larger:
14.35
- 8.60
------
5.75
2. The term with greater absolute value is -14.35 (negative).
3. Final result: -5.75
If the numbers have the same sign, add their absolute values and attach the common sign. If they have different signs, subtract the smaller absolute value from the larger absolute value and retain the sign of the number with the larger absolute value.
Multiplying Signed Fractions and Decimals
Multiplying rational numbers is straightforward because there is no requirement for common denominators or aligned decimal places. Simply multiply the magnitudes and apply the standard sign rules.
+-----------------------------------+-----------------------------------+
| Multiplying Fractions | Multiplying Decimals |
+-----------------------------------+-----------------------------------+
| 1. Convert mixed to improper | 1. Multiply ignoring decimals |
| 2. Cross-cancel common factors | 2. Count total decimal places |
| 3. Multiply numerators & denoms | 3. Place decimal in product |
| 4. Same signs -> +, Diff -> - | 4. Same signs -> +, Diff -> - |
+-----------------------------------+-----------------------------------+
Consider multiplying negative four-fifths by negative five-eighths. Both factors are negative, so the product will be positive. Multiplying the numerators four times five gives twenty, and denominators five times eight gives forty. Simplifying twenty-fortieths yields positive one-half. Cross-canceling before multiplying simplifies the arithmetic even further: the fives cancel to one, and four over eight reduces to one over two.
Multiplying Decimals Example: (-2.4) * 0.15
1. Multiply 24 * 15 = 360
2. Count decimal places: 2.4 has 1 place, 0.15 has 2 places (total = 3 places)
3. Position decimal: 0.360 = 0.36
4. Determine sign: (-) * (+) = (-)
Result: -0.36
Dividing Signed Fractions and Decimals
Division is the inverse of multiplication. To divide by a fraction, multiply by its multiplicative inverse, or reciprocal. The reciprocal of a fraction a/b is b/a. The sign of the reciprocal remains identical to the sign of the original number.
DIVIDING FRACTIONS: INVERT AND MULTIPLY
(a / b) / (c / d) = (a / b) * (d / c)
Consider negative seven-ninths divided by positive two-thirds. Invert the divisor to obtain its reciprocal, positive three-halves. Now multiply: negative seven-ninths times positive three-halves. Cross-canceling three and nine leaves three in the denominator. The product of numerators is negative seven times one, which is negative seven. The product of denominators is three times two, which is six. The resulting quotient is negative seven-sixths, or negative one and one-sixth.
Example: (-7/9) / (2/3) = (-7/9) * (3/2) = (-7 * 1) / (3 * 2) = -7/6
When dividing decimals, shift the decimal point in the divisor to make it an integer, shift the decimal point in the dividend by the exact same number of places to the right, and then perform long division.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Evaluate negative five-eighths minus negative three-fourths, expressing your final answer in simplest fractional form.
Exercise 2. A stock index begins the trading day at a price of forty-two dollars and fifty cents. Over three successive hours, its price changes by negative one dollar and forty cents, positive seventy-five cents, and negative two dollars and ten cents. Calculate the stock's closing price.
Exercise 3. Compute the product of negative two and one-third multiplied by positive one and five-sevenths.
Exercise 4. Evaluate the quotient of negative four point eight divided by negative zero point twelve.
Solutions
Solution to Exercise 1. First rewrite the subtraction as adding the opposite: negative five-eighths plus positive three-fourths. Find the least common denominator, which is eight. Rewrite positive three-fourths as positive six-eighths. Now combine the numerators: negative five plus six equals positive one. The result is positive one-eighth.
Solution to Exercise 2. Model the price changes as an addition expression: forty-two point fifty plus negative one point forty plus zero point seventy-five plus negative two point ten. Group the negative price drops: negative one point forty plus negative two point ten equals negative three point fifty. Now add the positive values: forty-two point fifty plus zero point seventy-five equals forty-three point twenty-five. Finally, add forty-three point twenty-five and negative three point fifty by subtracting: forty-three point twenty-five minus three point fifty equals thirty-nine point seventy-five. The closing price of the stock is thirty-nine dollars and seventy-five cents.
Solution to Exercise 3. Convert both mixed numbers into improper fractions. Negative two and one-third becomes negative seven-thirds. Positive one and five-sevenths becomes positive twelve-sevenths. Now multiply negative seven-thirds by positive twelve-sevenths. Cross-cancel the factor of seven in both the numerator and denominator to one. In the other diagonal, divide twelve by three to obtain four in the numerator and one in the denominator. The expression becomes negative one times four over one times one, which equals negative four.
Solution to Exercise 4. Both dividend and divisor have negative signs, so the quotient will be positive. To divide four point eight by zero point twelve, eliminate the decimal in the divisor by multiplying both divisor and dividend by one hundred (moving the decimal point two places to the right). This transforms the problem into four hundred eighty divided by twelve. Twelve goes into forty-eight four times, so twelve goes into four hundred eighty forty times. The final quotient is positive forty.