Comprehensive Study Guide: Operations with Rational Numbers and Fractions

Converting Fractions and Mixed Numbers to Decimals

• Fundamental Conversion Principle:

  • To convert any fraction or mixed number into a decimal, divide the numerator by the denominator.
  • Continue dividing until the remainder becomes zero, producing a terminating decimal, or until a specific sequence of digits begins repeating infinitely, producing a repeating decimal.
  • A repeating decimal uses a horizontal bar placed directly over the repeating digit or repeating block of digits.

• Step-by-Step Procedure for Terminating Decimals (Example 1):

  • Problem statement: Write $\frac{3}{4}$ as a decimal.
  • Step 1: Set up long division by dividing $3$ by $4$, written as $4\overline{)3.00}$.
  • Step 2: Place the decimal point in the quotient directly above the decimal point in the dividend.
  • Step 3: Divide $30$ tenths by $4$, yielding $0.7$ with a remainder: $4 \times 7 = 28$, and $30 - 28 = 2$.
  • Step 4: Bring down the next zero to make $20$ hundredths. Divide $20$ by $4$, yielding $5$: $4 \times 5 = 20$, and $20 - 20 = 0$.
  • Step 5: Because the remainder is zero, the division terminates: $\frac{3}{4} = 0.75$.

• Practice Exercises: Writing Fractions and Mixed Numbers as Decimals:

  • Exercise 1: $\frac{3}{10}$
  • Exercise 2: $\frac{2}{5}$
  • Exercise 3: $-\frac{5}{6}$
  • Exercise 4: $-7\frac{4}{9}$
  • Exercise 5: $\frac{5}{8}$
  • Exercise 6: $1\frac{4}{15}$

• Applied Real-World Word Problems:

  • Exercise 13: Antoine is cutting a $5\frac{5}{16}$-inch board for a project. Write $5\frac{5}{16}$ as a decimal.
  • Exercise 14: A basketball player successfully made $21$ out of $39$ free throw attempts. To the nearest thousandth, determine what part of the time he was successful in making his free throws.

Comparing and Ordering Rational Numbers

• Comparison Strategy:

  • To compare two rational numbers, express both numbers in the same numerical form (typically as decimals).
  • Compare corresponding place values starting from the greatest place value (left) and moving to the right until digits differ.
  • Insert the appropriate inequality or equality symbol: less than ($<$), greater than ($>$), or equal to ($=$).

• Step-by-Step Procedure for Comparing Rational Numbers (Example 2):

  • Problem statement: Replace $\bullet$ with $<$, $>$, or $=$ to make $\frac{4}{5} \bullet 0.75$ a true sentence.
  • Step 1: State the sentence: $\frac{4}{5} \bullet 0.75$.
  • Step 2: Convert the fraction $\frac{4}{5}$ to an equivalent decimal by dividing $4$ by $5$, which gives $0.8$.
  • Step 3: Rewrite the sentence using decimals: $0.8 \bullet 0.75$.
  • Step 4: Compare digits starting at the tenths place: $8 > 7$.
  • Step 5: Conclude that $0.8 > 0.75$, which makes $\frac{4}{5} > 0.75$ true.

• Practice Exercises: Comparing Rational Numbers:

  • Exercise 7: $\frac{3}{7} \bullet \frac{4}{9}$
  • Exercise 8: $-\frac{5}{8} \bullet -\frac{3}{5}$
  • Exercise 9: $2\frac{1}{2} \bullet 2\frac{5}{12}$
  • Exercise 10: $\frac{5}{8} \bullet 0.625$
  • Exercise 11: $4.37 \bullet 4\frac{19}{50}$
  • Exercise 12: $-2.54 \bullet -2\frac{27}{50}$

Rational Numbers: Converting Decimals to Fractions and Mixed Numbers

• Converting Terminating Decimals to Fractions in Simplest Form (Example 3):

  • Problem statement: Write $1.25$ as a fraction in simplest form.
  • Step 1: Identify the place value of the decimal portion. The number $1.25$ represents $1$ and $25$ hundredths.
  • Step 2: Express the decimal as a mixed number: $1.25 = 1\frac{25}{100}$.
  • Step 3: Simplify the fractional part by finding the Greatest Common Factor (GCF) of the numerator and the denominator. The GCF of $25$ and $100$ is $25$.
  • Step 4: Divide both the numerator and the denominator by their GCF: $\frac{25 \div 25}{100 \div 25} = \frac{1}{4}$.
  • Step 5: Combine with the whole number part to write in simplest form: $1\frac{1}{4}$.

• Converting Repeating Decimals to Fractions Using Algebraic Equations (Example 4):

  • Problem statement: Write $0.\overline{7}$ as a fraction in simplest form.
  • Step 1: Assign a variable to the repeating decimal: $N = 0.777\dots$
  • Step 2: Determine the number of repeating digits under the bar. Because exactly $1$ digit repeats, multiply each side of the equation by $10^1 = 10$:         $10N = 10(0.777\dots)$$10N = 7.777\dots$
  • Step 3: Subtract the original equation ($N = 0.777\dots$) from the multiplied equation ($10N = 7.777\dots$) to eliminate the infinitely repeating decimal tail:         $10N - N = 7.777\dots - 0.777\dots$
  • Step 4: Simplify the resulting algebraic expression:         $9N = 7$
  • Step 5: Divide each side by the coefficient $9$ to solve for $N$:         $N = \frac{7}{9}$

• Practice Exercises: Converting Decimals to Fractions or Mixed Numbers in Simplest Form:

  • Exercise 15: $2.08$
  • Exercise 16: $-0.45$
  • Exercise 17: $0.875$
  • Exercise 18: $-0.56$
  • Exercise 19: $0.\overline{1}$
  • Exercise 20: $-2.\overline{03}$
  • Exercise 21: $0.\overline{5}$
  • Exercise 22: $10.\overline{27}$

• Classification of Real Numbers into Number Sets:

  • Classify each given value by identifying all mathematical number sets to which it belongs (such as Natural numbers, Whole numbers, Integers, Rational numbers, or Irrational numbers):
    • Exercise 23: $-4$
    • Exercise 24: $3\frac{1}{3}$
    • Exercise 25: $1.151551555\dots$
    • Exercise 26: $-0.\overline{67}$

• Applied Word Problems on Rational Number Conversions:

  • Exercise 27: Suzanne practiced playing the piano for $1.\overline{6}$ hours after school. Write $1.\overline{6}$ as a mixed number in simplest form.
  • Exercise 28: James rode his motorbike for $10.4$ miles in a competition. Write $10.4$ as a mixed number in simplest form.

Adding and Subtracting Like Fractions

• Rules for Like Fractions:

  • Like fractions share identical denominators.
  • To add or subtract like fractions, add or subtract their numerators while retaining the common denominator.
  • Subtracting a negative fraction is mathematically equivalent to adding its positive opposite: $a - (-b) = a + b$.
  • Always simplify the final fraction to lowest terms or write as a mixed number.

• Step-by-Step Procedure for Subtracting Negative Like Fractions (Example 10):

  • Problem statement: Find $\frac{3}{4} - \left(-\frac{3}{4}\right)$. Write in simplest form.
  • Step 1: Rewrite subtraction of a negative value as addition of a positive value:         $\frac{3}{4} - \left(-\frac{3}{4}\right) = \frac{3}{4} + \frac{3}{4}$
  • Step 2: Denominators are the same, so add numerators over the common denominator:         $= \frac{3 + 3}{4}$
  • Step 3: Simplify the numerator:         $= \frac{6}{4}$
  • Step 4: Reduce to simplest form by dividing numerator and denominator by $2$ and converting to a mixed number:         $= 1\frac{1}{2}$

• Step-by-Step Procedure for Subtracting Mixed Numbers with Like Denominators (Example 11):

  • Problem statement: Find $5\frac{7}{8} - 8\frac{3}{8}$. Write in simplest form.
  • Step 1: Convert both mixed numbers into improper fractions:         $5\frac{7}{8} = \frac{5 \times 8 + 7}{8} = \frac{47}{8}$$8\frac{3}{8} = \frac{8 \times 8 + 3}{8} = \frac{67}{8}$$5\frac{7}{8} - 8\frac{3}{8} = \frac{47}{8} - \frac{67}{8}$
  • Step 2: Subtract the numerators over the shared denominator:         $= \frac{47 - 67}{8}$
  • Step 3: Simplify the numerator calculation:         $= \frac{-20}{8}$
  • Step 4: Divide numerator and denominator by their common factor of $4$ and express as an improper fraction or mixed number in simplest form:         $= -\frac{5}{2}\text{ or } -2\frac{1}{2}$

• Practice Exercises: Operations on Like Fractions:

  • Exercise 57: $\frac{8}{15} + \left(-\frac{2}{15}\right)$
  • Exercise 58: $\frac{6}{12} - \frac{11}{12}$
  • Exercise 59: $\frac{3}{7} - \left(-\frac{2}{7}\right)$
  • Exercise 60: $-\frac{1}{3} - \left(-\frac{1}{3}\right)$
  • Exercise 61: $2\frac{5}{12} - \left(-8\frac{7}{12}\right)$
  • Exercise 62: $5\frac{3}{7} + 2\frac{6}{7}$

• Applied Word Problems: Like Fractions:

  • Exercise 63: Samantha is going to walk $3\frac{5}{16}$ miles today and $2\frac{3}{16}$ miles tomorrow. What is the total distance she will walk?
  • Exercise 64: Last week, Douglas fed his puppy $10\frac{1}{4}$ cups of food. This week, the puppy will be fed an additional $1\frac{1}{4}$ cups of food. Find the total amount of food the puppy will be fed this week.
  • Exercise 65: Harry's sunflowers have grown to be $8\frac{1}{4}$ feet tall. Sonya's sunflowers are $6\frac{3}{4}$ feet tall. How much taller are Harry's flowers?
  • Exercise 66: Last month Clarissa read $41\frac{3}{8}$ books for the Read-a-thon. Mona read $27\frac{5}{8}$ books. How many more books did Clarissa read?

Adding and Subtracting Unlike Fractions

• Rules for Unlike Fractions:

  • Unlike fractions have different denominators.
  • Find the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the given denominators.
  • Rename each fraction by multiplying both numerator and denominator by the factor necessary to obtain the LCD.
  • Add or subtract the numerators while retaining the common LCD.
  • Simplify the resulting answer to simplest form.

• Step-by-Step Procedure for Adding Unlike Fractions (Example 12):

  • Problem statement: Find $-\frac{3}{8} + \frac{5}{6}$. Write in simplest form.
  • Step 1: Determine the LCD of the denominators $8$ and $6$. The least common multiple of $8$ and $6$ is $24$.
  • Step 2: Rename the fractions using the LCD of $24$:         $-\frac{3}{8} + \frac{5}{6} = -\frac{3 \cdot 3}{8 \cdot 3} + \frac{5 \cdot 4}{6 \cdot 4}$
  • Step 3: Simplify the fraction products:         $= -\frac{9}{24} + \frac{20}{24}$
  • Step 4: Combine the numerators over the common denominator:         $= \frac{-9 + 20}{24}$
  • Step 5: Perform numerator addition to reach simplest form:         $= \frac{11}{24}$

• Step-by-Step Procedure for Subtracting Mixed Numbers with Unlike Denominators and Renaming (Example 13):

  • Problem statement: Find $6\frac{5}{9} - 4\frac{11}{12}$. Write in simplest form.
  • Step 1: Identify the LCD of denominators $9$ and $12$, which is $36$.
  • Step 2: Rename the fractional parts using the LCD of $36$:         $6\frac{5}{9} - 4\frac{11}{12} = 6\frac{5 \times 4}{9 \times 4} - 4\frac{11 \times 3}{12 \times 3} = 6\frac{20}{36} - 4\frac{33}{36}$
  • Step 3: Compare fractional components. Because $\frac{20}{36}$ is less than $\frac{33}{36}$, regroup/rename $1$ whole from $6$ into thirty-sixths:         $6\frac{20}{36} = 5 + 1\frac{20}{36} = 5 + \frac{36}{36} + \frac{20}{36} = 5\frac{56}{36}$
  • Step 4: Rewrite the expression with the renamed mixed number:         $= 5\frac{56}{36} - 4\frac{33}{36}$
  • Step 5: Subtract the whole numbers, then subtract the numerators of the fractions:         $(5 - 4) + \left(\frac{56 - 33}{36}\right) = 1\frac{23}{36}$

• Practice Exercises: Operations on Unlike Fractions:

  • Exercise 67: $\frac{2}{5} + \frac{1}{15}$
  • Exercise 68: $-3\frac{5}{6} - 2\frac{1}{2}$
  • Exercise 69: $\frac{4}{7} + \left(-1\frac{1}{3}\right)$
  • Exercise 70: $\frac{3}{10} - \left(-\frac{1}{8}\right)$
  • Exercise 71: $25\frac{1}{3} - 14\frac{2}{5}$
  • Exercise 72: $7\frac{3}{4} + 1\frac{3}{8}$
  • Exercise 73: $-\frac{5}{9} - 3\frac{2}{3}$
  • Exercise 74: $-4\frac{1}{6} + \frac{3}{4}$

• Applied Word Problems: Unlike Fractions and Mixed Representations:

  • Exercise 75: Monica needs $2\frac{3}{4}$ cups of flour for a batch of cookies and $3\frac{1}{3}$ cups of flour for a dozen muffins. How many cups of flour does Monica need altogether?
  • Exercise 76: Dane and his family drove $357.9$ miles in one day. If their trip is a total of $524\frac{3}{4}$ miles, how much farther do they need to drive?
  • Exercise 77: Ricardo swam $75.5$ meters in the school pool. Helen swam $93\frac{3}{4}$ meters the same day. How much further did Helen swim that day?