Study guide: Rational Numbers as Fractions, Decimals, and Locations
Rational numbers can be represented as ratios of integers with nonzero denominator. Their decimal forms terminate or repeat, and negative quantities can be compared by their positions to the left of zero.
Ideas to explain in your own words
- A denominator built from factors two and five can be renamed as a power of ten. This connects exact fraction values to decimal place value without rounding. A decimal with three displayed places need not be approximate.
- A repeating decimal can be converted using two equations whose repeating tails match. Multiplying by ten shifts a one-digit repeating block so subtraction cancels the infinite tail. The dots mean the specified pattern repeats forever.
- Equivalent fractions put both quantities into the same-size parts. Signed numerators can then be added while the common denominator continues naming the size of each part. Add the signed numerators only after choosing a common denominator.
- For equal positive numerators, a smaller positive denominator creates a larger positive value. Negating both amounts reverses that ordering because the larger magnitude lies farther left of zero. Do not apply a positive-number comparison unchanged to negatives.
Useful vocabulary
- rational number
- A value expressible as an integer divided by a nonzero integer.
- repeating decimal
- A decimal with a digit block recurring without end.
- common denominator
- One denominator naming equal-size parts for several fractions.
Watch your reasoning
- A decimal with three displayed places need not be approximate. Distinguish an exact terminating value from a shortened decimal; dividing numerator by denominator preserves the fraction’s value.
- The dots mean the specified pattern repeats forever. A finite string such as 0.333 is different from 0.333…; do not replace an exact repeating fraction with an unexplained rounded value.
- Add the signed numerators only after choosing a common denominator. Adding denominators creates a different part size; keep the minus sign attached to its quantity during renaming.
A worked review example
Complete supplied guide examples: Rename an Eighth as a Decimal — Write 2/8 as an exact decimal. Candidate answers: (A) 0.025 (B) 0.25 (C) 0.5 (D) 4. Divide 2 by 8, or rename the denominator as 1000: 2/8=250/1000=0.25. Turn a Repeating Digit into a Fraction — Convert 0.111… (the digit 1 repeats forever) to a fraction. Candidate answers: (A) 1/10 (B) 1/9 (C) 1/99 (D) 2/9. Let x=0.111…. Then 10x=1.111…. Subtract x to get 9x=1, so x=1/9. This is an exact repeating decimal. Combine Signed Fractional Amounts — Compute −2/3 + 3/4 exactly. Candidate answers: (A) -2/7 (B) 1/12 (C) 1/7 (D) 17/12. Rename both fractions in twelfths: −8/12+9/12=(1)/12=1/12. The sign belongs to the numerator. Order Negative Fractions — Using numerator 2, which comparison of the supplied negative fractions is true? Candidate answers: (A) Negative fractions cannot be ordered (B) −2/3 < −2/4 (C) −2/3 = −2/4 (D) −2/3 > −2/4. 2/3>2/4 because thirds are larger than fourths for the same positive numerator. Negating reverses the order: −2/3 is farther left on the number line.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.