10. Comparing Fractions and Mixed Numbers

Session 10: Comparing Fractions and Mixed Numbers

Learning targets

• Compare fractions using benchmarks and common denominators.
• Use cross-products as a comparison method.
• Convert improper fractions and mixed numbers.
• Order positive and negative fractions.

Why this matters

Comparison should be based on value, not on whichever numerator or denominator looks larger. Multiple representations help learners choose an efficient method and explain the decision.

Core lesson

Fractions with equal denominators can be compared by numerators; fractions with equal positive numerators have smaller values when their denominators are larger. For unlike fractions, a common denominator creates equal-sized pieces. Benchmarks such as one-half and 1 may settle a comparison without exact conversion.

Cross-products compare positive fractions because they create equivalent numerators over a common product denominator. For \(\frac58\) and \(\frac23\), compare \(5\cdot3=15\) with \(2\cdot8=16\), so \(\frac58<\frac23\). Mixed numbers should be compared by whole parts first. With negative fractions, number-line order still rules: the value closer to zero is greater.

Foundations in depth

Fractions can be compared only when the part sizes or the number of parts are made compatible. With equal denominators, compare numerators. With equal numerators and positive denominators, the fraction with the smaller denominator is larger because each part is larger. For unlike fractions, use a common denominator, cross-products, decimals, or a benchmark.

Mixed numbers should be considered from the whole-number part first. If whole parts differ, the larger whole part determines the comparison. If they match, compare the fractional parts. Cross-products should be interpreted, not memorized: for a/b and c/d with positive denominators, ad and bc are the numerators produced when both fractions are rewritten over bd. Estimation with one-half or one catches reversed comparisons.

A reliable method

1. Use a benchmark to predict the result.
2. Create a common denominator, decimal form, or cross-product.
3. Compare the equivalent values.
4. Verify the order on a number line.

Ten GED®-style worked examples

Worked example 1

GED®-style problem. Compare \(\frac7{12}\) and \(\frac58\).

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Compare whole parts first, then use common parts, cross-products, or decimals.

Step 3 — Solve carefully. Cross-products are \(7\cdot8=56\) and \(5\cdot12=60\), so \(\frac7{12}<\frac58\).

Step 4 — Verify and state the answer. Check the order against benchmarks such as one-half and one.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 2

GED®-style problem. Order \(1\frac25\), \(\frac75\), and \(1.35\).

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Compare whole parts first, then use common parts, cross-products, or decimals.

Step 3 — Solve carefully. \(1\frac25=\frac75=1.4\), so \(1.35<1\frac25=\frac75\).

Step 4 — Verify and state the answer. Check the order against benchmarks such as one-half and one.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 3

GED®-style problem. Compare \(-\frac34\) and \(-\frac23\).

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Compare whole parts first, then use common parts, cross-products, or decimals.

Step 3 — Solve carefully. \(-0.75<-0.667\ldots\), so \(-\frac34< -\frac23\).

Step 4 — Verify and state the answer. Check the order against benchmarks such as one-half and one.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 4

GED®-style problem. Compare 7/10 and 2/3.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Use a common denominator or cross-products.

Step 3 — Solve carefully. A common denominator is 30: 7/10=21/30 and 2/3=20/30. Therefore 7/10>2/3.

Step 4 — Verify and state the answer. Decimals 0.7 and about 0.667 support the comparison.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 5

GED®-style problem. Order 3/8, 5/12, and 1/2 from least to greatest.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Use the least common denominator 24.

Step 3 — Solve carefully. 3/8=9/24, 5/12=10/24, and 1/2=12/24. Thus 3/8<5/12<1/2.

Step 4 — Verify and state the answer. All are at or below one-half, with increasing numerators over equal parts.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 6

GED®-style problem. Which is greater, 4 2/7 or 4 3/8?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Whole-number parts match, so compare only the fractions.

Step 3 — Solve carefully. Compare 2/7 and 3/8: 2×8=16 and 3×7=21. Since 21>16, 3/8>2/7. Therefore 4 3/8 is greater.

Step 4 — Verify and state the answer. Approximate fractions 0.286 and 0.375 confirm the result.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 7

GED®-style problem. A board is 5 5/6 ft long. Another is 5.8 ft. Which is longer?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Convert the fractional part to a decimal.

Step 3 — Solve carefully. 5/6≈0.833, so 5 5/6≈5.833 ft. Since 5.833>5.800, the first board is longer.

Step 4 — Verify and state the answer. The difference is about 0.033 ft, a small but positive amount.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 8

GED®-style problem. Place 9/10, 11/12, and 7/8 in decreasing order.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Compare each value's distance below 1.

Step 3 — Solve carefully. The gaps from 1 are 1/10=0.1, 1/12≈0.0833, and 1/8=0.125. The smallest gap gives the greatest fraction. Order: 11/12>9/10>7/8.

Step 4 — Verify and state the answer. Decimal approximations 0.917, 0.900, and 0.875 agree.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 9

GED®-style problem. A container is 13/16 full. Is it more or less than 80% full?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Convert 80% to 4/5 and compare fractions.

Step 3 — Solve carefully. Compare 13/16 and 4/5: 13×5=65 and 4×16=64. Since 65>64, 13/16 is slightly more than 80%.

Step 4 — Verify and state the answer. 13/16=0.8125=81.25%, which is 1.25 percentage points above 80%.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 10

GED®-style problem. Which fraction lies between 2/5 and 1/2: 3/8, 7/15, or 5/9?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Convert or compare each candidate with both boundaries.

Step 3 — Solve carefully. 2/5=0.4 and 1/2=0.5. Candidates: 3/8=0.375, 7/15≈0.467, 5/9≈0.556. Only 7/15 lies between 0.4 and 0.5.

Step 4 — Verify and state the answer. Cross-products confirm 2/5<7/15 and 7/15<1/2.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Adult-life and GED® applications

This session's reasoning applies to:

• selecting material sizes
• comparing completion rates
• ordering measurements

When the setting changes, keep the mathematical relationship and unit logic visible. Do not choose an operation from a keyword alone. First decide what the quantities mean and how they relate.

Common traps and repairs

• Choosing the larger denominator: Denominator size changes the unit piece.
• Using cross-products without keeping order: Match each cross-product to its original fraction.
• Ignoring negative order: Among negatives, the value closer to zero is greater.

Retrieval and mixed practice

Retrieve equivalent fractions from Session 9 and place \(1/4\), \(1/2\), \(3/4\), and \(5/4\) on a number line.

After recalling an answer, compare it with the worked reasoning and correct the explanation, not only the final number. Then mix this session's skill with at least one earlier topic so method selection becomes part of the practice.

Session check

Exit reflection

Complete these two sentences: The main decision in this session is... and I can check my answer by... Keep both statements specific enough to use on a new problem.