Convert Number Forms — Convert between fractions, decimals, and percentages. - TEAS Mathematics

SECTION 1

Why This Skill Matters

The ability to represent the same quantity in multiple forms — as a fraction, a decimal, or a percentage — is a core quantitative skill tested directly on the TEAS. In health science and nursing programs, you will routinely encounter all three forms: medication dosages expressed as fractions, lab values reported as decimals, and survey results stated as percentages. Being able to move fluidly among these representations lets you choose the most convenient form for any given calculation and avoid errors caused by mismatched notation.

The central question is deceptively simple: given a quantity expressed in one form, how do we efficiently and accurately rewrite it in another form? On the TEAS, this skill undergirds dosage calculations, data interpretation, and proportional reasoning — making it a foundational competency that merits both conceptual understanding and procedural fluency.

SECTION 2

Core Principles & Definitions

At their core, fractions, decimals, and percentages are three equivalent representations of rational numbers. A fraction expresses a ratio of two integers; a decimal expresses the same value using base-10 positional notation; and a percentage scales the decimal form by a factor of 100 for intuitive comparison. Mastery of conversion requires internalizing four foundational principles that govern the relationships among these forms.

Equivalence Principle

A fraction a/b, its decimal quotient a ÷ b, and its percentage (a ÷ b) × 100% all denote the same position on the number line. Conversion changes notation, not value.

Division Underpins All Forms

Every fraction is an implied division. Performing that division yields the decimal; multiplying the decimal by 100 yields the percentage. Division is the universal gateway operation.

Place Value as Denominator

Each decimal digit occupies a place whose value is a power of 10. The decimal 0.375 is 375/1000, making the denominator implicit in the positional notation — a key insight for decimal-to-fraction conversion.

Percentage as a Centesimal Ratio

Percent literally means 'per hundred.' Thus 42% = 42/100 = 0.42. This fixed denominator of 100 is what makes percentages so convenient for direct comparison across different quantities.

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KEY TAKEAWAY

Think of fractions, decimals, and percentages as three different languages describing the same address on the number line. Converting between them is akin to translating a sentence from French to English to Spanish: the meaning is invariant, but the syntax changes. On the TEAS, choosing the most convenient 'language' for a given problem — say, percentages for comparison or decimals for calculation — is often the key to both speed and accuracy.

SECTION 3

Visual Explanation — The Conversion Triangle

The Conversion Triangle illustrates the six directional pathways among fractions, decimals, and percentages. The solid green arrow from Fraction to Decimal represents performing the division a ÷ b. The solid amber arrow from Fraction to Percent adds a multiplication by 100. The bottom cyan arrow from Decimal to Percent is simply multiplying by 100, and the reverse violet dashed arrow divides by 100. Dashed arrows indicate reverse operations: from Decimal back to Fraction, read the place value and simplify; from Percent back to Fraction, divide by 100 and simplify.

The Conversion Triangle above encapsulates the entire algorithmic framework in a single image. Notice that division is the fundamental operation linking fractions to decimals, while multiplication or division by 100 mediates the decimal–percentage transition. Every conversion can be decomposed into at most two of these elementary steps, which is why the triangle has no 'shortcut' edges that bypass a vertex — the operations are intrinsically sequential. For graduate-level test preparation, internalizing this triangle allows you to reconstruct any conversion procedure from first principles, even under time pressure.

SECTION 4

Mathematical Framework

The six conversion pathways in the triangle can each be expressed as a concise formula. These equations are not merely recipes — they follow directly from the definitions of decimal positional notation and the centesimal interpretation of percentages. Internalizing them as algebraic identities, rather than rote procedures, enables flexible application across novel problem contexts.

FRACTION → DECIMAL

d = a ÷ b

Where a is the numerator, b is the denominator (b ≠ 0), and d is the resulting decimal. If b's prime factorization contains only 2s and 5s, the decimal terminates; otherwise, it repeats.

DECIMAL → FRACTION

d = n / 10ᵏ, then simplify by GCD(n, 10ᵏ)

Read d as n ten-thousandths (or hundredths, thousandths, etc.), where k is the number of decimal places and n is the integer formed by removing the decimal point. Divide numerator and denominator by their greatest common divisor to simplify.

DECIMAL → PERCENTAGE

P = d × 100%

Multiply the decimal by 100 and append the percent sign. Equivalently, shift the decimal point two places to the right.

PERCENTAGE → DECIMAL

d = P / 100

Remove the percent sign and divide by 100, or equivalently shift the decimal point two places to the left. For example, 7.5% = 0.075.

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Repeating Decimals

When converting fractions like 1/3 or 5/6 to decimals, the result is a repeating decimal (0.333… or 0.8333…). On the TEAS, you may need to round to a specified number of decimal places or recognize the repeating pattern. To convert a repeating decimal back to a fraction, use the algebraic method: let x = 0.333…, then 10x = 3.333…, so 9x = 3, giving x = 3/9 = 1/3.

SECTION 5

Common Equivalences & Classification

While every conversion can be computed from first principles, speed on the TEAS is substantially improved by memorizing a set of benchmark equivalences. The table below catalogues the most frequently tested conversions, organized by denominator family. Recognizing these instantly frees cognitive resources for the higher-order reasoning the exam demands.

Fraction	Decimal	Percentage	Notes
1/2	0.5	50%	Halves family
1/4	0.25	25%	Quarters family
3/4	0.75	75%	Quarters family
1/5	0.2	20%	Fifths family
1/3	0.333…	33.3̄%	Repeating; thirds family
2/3	0.666…	66.6̄%	Repeating; thirds family
1/8	0.125	12.5%	Eighths family
3/8	0.375	37.5%	Eighths family
1/10	0.1	10%	Tenths family; trivial conversion
1/6	0.1666…	16.6̄%	Repeating; sixths family

Benchmark Fraction–Decimal–Percentage Equivalences

The upper portion of this diagram places key benchmark values on a 0-to-1 number line, displaying each value simultaneously as a fraction, decimal, and percentage. The lower panel classifies fractions into terminating or repeating decimals based on the prime factorization of the denominator — a distinction critical for anticipating whether a conversion yields a clean result or requires rounding.

The classification of decimal outcomes as terminating or repeating has direct practical consequences. When a TEAS item asks you to express 5/6 as a decimal, you should immediately recognize that 6 = 2 × 3, and the presence of the factor 3 guarantees a repeating decimal. This awareness prevents futile attempts at exact computation and signals that the answer will likely involve a rounded or repeating notation.

SECTION 6

Worked Example — Multi-Step Conversion

Consider the following problem, representative of TEAS-level complexity: A laboratory reports that 7 out of 16 blood samples tested positive. Express this result as a decimal and as a percentage, rounded to one decimal place.

Converting 7/16 to Decimal and Percentage

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Step 1 — Identify the Fraction

The problem states 7 out of 16 samples, which directly yields the fraction 7/16. Verify that 16 = 2⁴, so the denominator consists solely of factors of 2, guaranteeing a terminating decimal.

Fraction: 7/16
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Step 2 — Perform the Division

Compute 7 ÷ 16. Using long division or recognizing that 7/16 = 7 × (1/16) = 7 × 0.0625: we get 7 × 0.0625 = 0.4375. Alternatively, one may use the equivalence 1/16 = 625/10000 and multiply: 7 × 625 = 4375, so 7/16 = 4375/10000 = 0.4375.

Decimal: 0.4375
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Step 3 — Convert Decimal to Percentage

Apply the formula P = d × 100%: multiply 0.4375 × 100 = 43.75%. Since the problem requests one decimal place, this is already expressed to the appropriate precision; no rounding is needed.

Percentage: 43.75% ≈ 43.8% (rounded to one decimal place)
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Step 4 — Verify by Reverse Conversion

As a check, convert 43.75% back to a fraction: 43.75/100 = 4375/10000. Simplify by dividing numerator and denominator by 625: 4375 ÷ 625 = 7 and 10000 ÷ 625 = 16, recovering 7/16. The round-trip conversion confirms the result.

Verified: 43.75% = 7/16 ✓

✓

VERIFICATION STRATEGY

Always verify a conversion by performing the reverse operation. This round-trip check takes only seconds but catches arithmetic errors that could cost points on the TEAS. Think of it as a checksumming protocol — the same logic used in data transmission to ensure a message arrives intact. If your reverse conversion does not yield the original value, an error occurred somewhere in the chain.

SECTION 7

Strengths & Limitations of Each Form

Each number form has contexts in which it excels and contexts in which it is cumbersome. The strategic choice of representation can dramatically simplify problem-solving, particularly under timed conditions. The table below summarizes the comparative strengths and limitations.

Criterion	Fraction	Decimal	Percentage
Exact representation	Always exact (e.g., 1/3)	May require rounding for repeaters	May require rounding (e.g., 33.3̄%)
Comparison ease	Requires common denominator	Directly comparable via place value	Directly comparable; intuitive scale
Arithmetic ease	Multiplication is straightforward; addition requires LCD	All four operations use standard algorithms	Requires conversion to decimal for computation
Communicative clarity	Less intuitive for general audience	Good for scientific/technical context	Best for lay communication and statistics
TEAS use cases	Dosage ratios, proportions	Calculations, metric conversions	Data interpretation, probability

Comparative Analysis of Number Forms

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STRATEGIC FORM SELECTION

The optimal form depends on the operation. Use fractions for exact proportional reasoning (especially when denominators share factors), decimals for computation (addition, subtraction, multiplication with a calculator), and percentages for quick comparison and communication. On the TEAS, scanning the answer choices first often reveals which form the test expects, guiding your conversion strategy.

SECTION 8

Connections to Advanced Quantitative Reasoning

The ability to fluidly convert among number forms is not merely an isolated arithmetic skill — it serves as the foundational layer for more sophisticated quantitative reasoning encountered in graduate-level coursework and professional practice. The table below maps each TEAS conversion skill to its advanced counterpart, illustrating how this lesson's concepts scale into domains such as pharmacology, biostatistics, and clinical research.

TEAS-Level Skill	Advanced Application	Example Context
Fraction → Decimal	Dosage calculation from concentration ratios	A 1:1000 epinephrine solution = 0.001 g/mL
Decimal → Percentage	Interpreting effect sizes, confidence levels	A p-value of 0.03 represents a 3% probability
Percentage → Fraction	Simplifying compound interest and rate calculations	A 4.5% annual rate = 9/200 in exact form
Repeating decimal recognition	Identifying rational vs. irrational results in measurement	Distinguishing measurement error from true irrationality
Benchmark memorization	Mental estimation in clinical decision-making	Quickly estimating that 3/8 of a 400 mg dose ≈ 150 mg

TEAS Skills and Their Advanced Counterparts

In graduate programs — particularly in nursing, health sciences, and public health — you will encounter these conversion skills embedded within multi-step applied problems. The TEAS assesses whether you have automated the basic conversion operations to the point where they can be deployed as subconscious subroutines within larger problem-solving frameworks, rather than requiring dedicated cognitive effort at each step.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the fraction 7/12 must produce a repeating decimal rather than a terminating one. What property of the denominator determines this outcome?

Show answer

A fraction in lowest terms produces a terminating decimal if and only if the denominator's prime factorization contains no primes other than 2 and 5. The denominator 12 factors as 2² × 3, and the presence of the prime factor 3 guarantees a repeating decimal. Specifically, 7 ÷ 12 = 0.58333…, where the digit 3 repeats indefinitely. This principle generalizes: any denominator containing a factor of 3, 7, 11, 13, or any prime other than 2 or 5 will produce a non-terminating, repeating decimal representation.

Question 2: BASIC CALCULATION

Convert 0.875 to a fraction in lowest terms and then express it as a percentage.

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The decimal 0.875 has three decimal places, so it equals 875/1000. To simplify, find GCD(875, 1000). Since 875 = 5³ × 7 and 1000 = 2³ × 5³, the GCD is 5³ = 125. Dividing: 875 ÷ 125 = 7 and 1000 ÷ 125 = 8, yielding the fraction 7/8. To convert to a percentage, multiply the decimal by 100: 0.875 × 100 = 87.5%. Alternatively, recognize that 7/8 is a benchmark value from the eighths family, confirming the result.

Question 3: INTERMEDIATE

A patient's chart indicates that 62.5% of their daily fluid intake should be oral. Express this as a fraction in lowest terms, then determine what decimal this fraction equals. If the total required intake is 2400 mL, how many milliliters should be oral?

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Begin by converting 62.5% to a fraction: 62.5/100 = 625/1000. Simplify by noting GCD(625, 1000) = 125: 625 ÷ 125 = 5 and 1000 ÷ 125 = 8, yielding 5/8. The decimal equivalent is 5 ÷ 8 = 0.625, which is consistent with the original percentage (0.625 × 100 = 62.5%). For the applied portion: 0.625 × 2400 mL = 1500 mL of oral intake. Alternatively, compute 5/8 × 2400 = (5 × 2400)/8 = 12000/8 = 1500 mL, confirming the answer through both decimal and fractional methods.

Question 4: APPLIED

A pharmacist needs to prepare a solution that is 0.45% saline. Express this concentration as a decimal and as a fraction in simplest form. If 500 mL of this solution is prepared, how many grams of saline are required?

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Converting 0.45% to a decimal: 0.45 ÷ 100 = 0.0045. Note the critical distinction: 0.45% is less than one percent, not 45%. As a fraction: 0.45/100 = 45/10000. Simplify by dividing by GCD(45, 10000) = 5: 45 ÷ 5 = 9 and 10000 ÷ 5 = 2000, yielding 9/2000. For the applied calculation: if 0.45% means 0.45 grams per 100 mL, then for 500 mL the required amount is 0.0045 × 500 = 2.25 grams. Equivalently, 9/2000 × 500 = 4500/2000 = 2.25 grams. This problem tests the common trap of confusing a percentage like 0.45% with the decimal 0.45.

Question 5: CRITICAL THINKING

A student claims that 0.1̄ (0.111…) = 1/9 and 0.9̄ (0.999…) = 9/9 = 1, which would mean 0.999… equals exactly 1, not merely 'close to' 1. Construct a rigorous algebraic argument to prove or disprove this claim, and discuss the implications for percentage conversion of repeating decimals.

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The student's claim is correct and can be proven algebraically. Let x = 0.999…. Then 10x = 9.999…. Subtracting: 10x − x = 9.999… − 0.999…, so 9x = 9, giving x = 1. This is not a paradox but a consequence of the completeness of the real numbers: two distinct decimal representations can denote the same real number. Since 1/9 = 0.111…, multiplying both sides by 9 yields 9/9 = 0.999… = 1, which is consistent. The implication for percentage conversion is profound: when we write 1/3 = 33.3̄%, the repeating notation is exact, not an approximation. The percentage 33.333…% is precisely equal to 1/3, and any finite truncation introduces an error whose magnitude depends on the number of retained digits. This understanding is essential for interpreting precision in clinical and scientific contexts.

SUMMARY

Lesson Summary

Summary

Fractions, decimals, and percentages are three equivalent representations of rational numbers, unified by the fundamental operation of division. To convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a percentage, multiply by 100. To convert a decimal to a fraction, read the place value as the denominator and simplify by the GCD. Whether a fraction yields a terminating or repeating decimal depends on whether the simplified denominator's prime factors are exclusively 2s and 5s.

For TEAS success, memorize benchmark equivalences (halves, quarters, thirds, fifths, eighths) to accelerate recognition, select the optimal form for each problem context (fractions for exact ratios, decimals for computation, percentages for comparison), and always verify conversions with a reverse operation. These skills serve as the quantitative substrate for dosage calculations, data interpretation, and proportional reasoning throughout graduate-level health science programs.