Real Numbers, Ratios, and Percents

Why Real-Number Fluency Matters on Praxis 5165

Number & Quantity and Algebra is weighted at 30% of Praxis Mathematics Content Knowledge (5165). Before you manipulate variables, ETS checks whether you understand how real numbers behave—comparing unlike forms, setting up ratios, and interpreting percents in context. A surprising number of items are lost not on advanced algebra but on a sloppy conversion between sqrt(5), 9/4, and 2.23.

The real numbers include all rationals (fractions, terminating or repeating decimals) and irrationals (sqrt(2), pi, non-repeating decimals). On the exam, you rarely need formal proofs; you need fast, accurate comparison and modeling.

The Real Number Hierarchy

Every real number has a location on the number line. Density means between any two reals there is another rational—useful when ETS asks which value lies between two given endpoints.

Comparing Unlike Forms

When options mix radicals, fractions, and decimals, pick one representation:

Worked Example 1 — Greatest value

Which is greatest: sqrt(5), 2.1, 9/4, 2.23?

• sqrt(5) ≈ 2.236 (because 2.2² = 4.84 and 2.25² = 5.0625)
• 2.1 = 2.10
• 9/4 = 2.25
• 2.23 = 2.23

9/4 is greatest. Praxis stems often place sqrt(5) near 2.24 to tempt you into picking the radical.

Worked Example 2 — Ordering negatives

Order from least to greatest: −3/5, −0.58, −sqrt(0.36), −0.55.

• −3/5 = −0.60
• −sqrt(0.36) = −0.60 (same as −3/5)
• −0.58 and −0.55 are larger (closer to zero)

Least: −0.60 (two equal values), then −0.58, then −0.55.

Worked Example 3 — Between which integers?

sqrt(50) lies between which consecutive integers?

sqrt(49) = 7 and sqrt(64) = 8, so sqrt(50) is between 7 and 8. Exact value ≈ 7.07.

Ratios and Proportions

A ratio compares two quantities with the same units. Write 3 teachers : 45 students as 3/45 = 1/15 teacher per student, or scale to 1 : 15.

A proportion states two ratios are equal: a/b = c/d. Cross-multiply when b and d are nonzero: ad = bc.

Worked Example 4 — Map scale

A map uses 1 inch = 12 miles. How many miles are represented by 3.5 inches?

Set up 1/12 = 3.5/x → x = 3.5 × 12 = 42 miles.

For unit rates, divide total by units: 280 miles in 4 hours → 70 mi/h.

Worked Example 5 — Recipe scaling

A recipe for 6 servings uses 2.5 cups flour. How much flour for 15 servings?

2.5/6 = x/15 → x = 2.5 × 15/6 = 6.25 cups.

Percents

Percent means per hundred. Convert: fraction × 100%, or move the decimal two places.

Worked Example 6 — Percent decrease

A textbook price drops from $80 to $68.

Change = (68 − 80)/80 = −12/80 = −0.15 → 15% decrease. The new price is 85% of the original.

Trap: A 20% discount followed by a 20% markup does not return to the original price because the second percent applies to a smaller base.

Worked Example 7 — Tax and tip

Meal $45, tax 8%, tip 18% on pre-tax subtotal.

Tax = 0.08 × 45 = $3.60. Tip = 0.18 × 45 = $8.10. Total = 45 + 3.60 + 8.10 = $56.70.

Worked Example 8 — Percent error

Measured length 24.2 cm, actual 25.0 cm.

Percent error = |24.2 − 25.0|/25.0 × 100% = 0.8/25 × 100% = 3.2%.

Scientific Notation and Orders of Magnitude

Large and small numbers appear in data-heavy stems. Write a × 10^n with 1 ≤ |a| < 10.

3,400,000 = 3.4 × 10^6.

Exam Strategy

• Convert before comparing; do not rely on gut feel for radicals.
• Label ratio units—student papers that invert teacher:student ratios are common in teaching scenarios.
• For percent increase/decrease, always divide by the original amount.
• When ETS shows four decimal approximations, estimate radicals with nearby perfect squares.
• Link to free Praxis Math 5165 practice questions for mixed number-sense drills.

Teaching-Scenario Watchpoints

If a student ranks sqrt(10) below 3 because "square roots are small," the misconception is treating sqrt as division. Counter: sqrt(9) = 3 and sqrt(16) = 4, so sqrt(10) > 3. Diagnose the comparison step before commenting on pedagogy.

Repeating Decimals and Exact Fractions

A rational number can be written as a fraction. Repeating decimals such as 0.333… equal 1/3, and 0.1666… equals 1/6. On Praxis items, converting to fractions often makes comparison exact.

Worked Example 9 — Repeating decimal

Write 0.454545… as a fraction. Let x = 0.454545…. Then 100x = 45.4545…, so 100x − x = 45 → x = 45/99 = 5/11.

Absolute Value and Distance on the Number Line

|a − b| is the distance between a and b. This idea reappears in absolute-value equations later in the chapter but starts here as number sense.

Distance between −4 and 3 is |−4 − 3| = 7.

Part-Part-Whole Ratio Models

When a stem gives a ratio of part to part, find the whole before answering percent questions. If boys:girls = 2:3 in a class of 30 students, boys = (2/5)(30) = 12. The denominator 5 is the sum of ratio parts, not the class size itself.