Perform Arithmetic Operations — Perform operations with whole numbers, decimals, and fractions. - TEAS Mathematics

SECTION 1

Historical Context & Motivation

The four fundamental arithmetic operations — addition, subtraction, multiplication, and division — form the bedrock upon which all quantitative reasoning is constructed. Long before the development of formal algebra, geometry, or calculus, human civilizations required reliable methods for counting livestock, apportioning land, and facilitating trade. The history of arithmetic is therefore inseparable from the history of civilization itself, and understanding how these operations evolved provides insight into why certain algorithms work the way they do. For graduate-level test takers preparing for the TEAS, revisiting these fundamentals is not merely remedial; it ensures the kind of fluency and speed required when more complex nursing-related calculations — dosage computations, fluid ratios, nutritional percentages — depend on error-free arithmetic under timed conditions.

c. 3000 BCE • Babylonian Numeration

The Babylonians developed a base-60 positional number system, enabling sophisticated multiplication and division on clay tablets. Their approach to fractions — expressed as sums of unit fractions of 60 — foreshadowed modern decimal expansions.

c. 300 BCE • Euclid's Elements

Euclid formalized the properties of whole numbers, including divisibility, prime factorization, and the Euclidean algorithm for finding greatest common divisors — a procedure still essential when simplifying fractions.

c. 825 CE • Al-Khwārizmī's Arithmetic

The Persian mathematician al-Khwārizmī introduced Hindu-Arabic numerals and positional notation to the Islamic world; his treatise on arithmetic gave us the word 'algorithm' and standardized the decimal place-value system we use today.

1585 • Stevin's Decimal Fractions

Simon Stevin published 'De Thiende,' demonstrating that decimal fractions could replace common fractions in practical computation, transforming commerce and science across Europe.

Modern Era • Computational Fluency in Healthcare

Contemporary nursing and allied health fields demand rapid, precise arithmetic — from medication dosage calculations involving fractions and decimals to intravenous drip rate conversions — making operational mastery a clinical imperative.

The central question this lesson addresses is deceptively simple: how do we perform addition, subtraction, multiplication, and division across the three principal numeric representations — whole numbers, decimals, and fractions — with consistent accuracy and efficiency? By the end of this lesson you will have the procedural fluency and conceptual clarity needed to handle any arithmetic problem the TEAS exam can present.

SECTION 2

Core Principles & Definitions

Before engaging with specific algorithms, it is important to solidify the conceptual framework that governs all arithmetic operations. The four operations — addition, subtraction, multiplication, and division — are interconnected through inverse relationships and obey a consistent set of algebraic properties regardless of whether the operands are whole numbers, decimals, or fractions. Understanding these properties allows you to verify results, simplify computations, and recognize equivalent expressions on the TEAS.

Commutative Property

For addition and multiplication, the order of operands does not affect the result: a + b = b + a and a × b = b × a. This property does not hold for subtraction or division.

Associative Property

Grouping of operands can be rearranged in addition and multiplication without changing the outcome: (a + b) + c = a + (b + c). This is critical when regrouping numbers mentally to simplify computation.

Distributive Property

Multiplication distributes over addition: a × (b + c) = a × b + a × c. This property bridges multiplication and addition and is essential for mental math strategies.

Identity & Inverse Elements

The additive identity is 0 (a + 0 = a) and the multiplicative identity is 1 (a × 1 = a). Every number has an additive inverse (−a) and, if nonzero, a multiplicative inverse (1/a).

Place Value & Equivalence

Whole numbers, decimals, and fractions are different notational systems for the same set of rational numbers. Converting between them — e.g., 0.75 = 3/4 = 75/100 — is often the first step in solving mixed-format problems.

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KEY TAKEAWAY

Think of arithmetic properties as the 'grammar rules' of mathematics: just as subject-verb agreement ensures a sentence is understood regardless of vocabulary, the commutative, associative, and distributive properties ensure that computations yield the same result regardless of how you organize the work. Mastering these rules lets you choose the most efficient route through any calculation — an advantage under timed conditions.

SECTION 3

Visual Explanation — The Number Line & Place Value

A powerful way to unify whole numbers, decimals, and fractions is to view them all as points on the number line. The diagram below illustrates how 0.75, 3/4, and 75/100 occupy exactly the same position, while whole numbers sit at regular integer intervals. Arithmetic operations correspond to geometric movements along this line: addition moves right, subtraction moves left, and multiplication scales distances from the origin.

The number line shows how whole numbers (0, 1, 2), decimals (0.5, 0.75, 1.5), and fractions (1/2, 3/4, 3/2) all occupy specific positions. The dashed arcs illustrate addition as rightward movement along the line. The inset box decomposes 0.75 by place value.

This unified view reinforces a critical insight: the procedures for adding, subtracting, multiplying, and dividing are not three separate skill sets but rather variations on the same positional-value logic. When you add 0.75 + 0.25, you are performing the same column-wise regrouping as 75 + 25, but with a decimal point anchoring the place values. Similarly, adding 3/4 + 1/4 requires a common denominator — effectively aligning 'place values' in the fractional world. The visual helps clarify that mastery of one representation naturally reinforces competence in the others.

SECTION 4

Mathematical Framework — Operations in Detail

Whole Number Operations

Operations with whole numbers rely on the standard algorithms taught in elementary education, but at the graduate-admission level the emphasis shifts to speed and accuracy. Carrying in addition, borrowing in subtraction, and careful alignment in long multiplication and division should be second nature. The formal definitions are as follows.

ADDITION & SUBTRACTION

a + b = S and a − b = D, where a ≥ b for non-negative D

S = sum, D = difference. When subtracting, regroup (borrow) when a digit in the minuend is smaller than the corresponding digit in the subtrahend.

MULTIPLICATION

a × b = P (distributive form: a × (b₁ + b₂) = a × b₁ + a × b₂)

P = product. Multi-digit multiplication decomposes one factor by place value, multiplies each partial product, and then sums the results.

DIVISION

a ÷ b = Q remainder R, where a = b × Q + R and 0 ≤ R < b

Q = quotient, R = remainder, a = dividend, b = divisor. When R = 0, b divides a evenly.

Decimal Operations

Decimal arithmetic extends whole-number algorithms with one key rule: align the decimal points for addition and subtraction, and count total decimal places for multiplication. In addition and subtraction, pad with trailing zeros as necessary so both operands have the same number of decimal places. For multiplication, multiply as if both numbers were whole numbers, then insert the decimal point so the product has a number of decimal places equal to the sum of decimal places in both factors. In division, move the decimal point in the divisor to make it a whole number, shift the decimal in the dividend by the same count, and then proceed with long division.

DECIMAL MULTIPLICATION RULE

If a has m decimal places and b has n decimal places, then a × b has m + n decimal places.

Example: 1.25 (2 places) × 0.4 (1 place) = 0.500 (3 places) = 0.5.

Fraction Operations

Fractions require distinct algorithms for each operation. For addition and subtraction, the operands must share a least common denominator (LCD). Multiplication is the most straightforward: multiply numerators together and denominators together. Division requires the reciprocal of the divisor — often phrased as 'invert and multiply.' Always simplify the result by dividing the numerator and denominator by their greatest common divisor (GCD).

FRACTION ADDITION / SUBTRACTION

a/b ± c/d = (a × d ± c × b) / (b × d), then simplify

This cross-multiplication formula guarantees a common denominator. When b and d share a common factor, using the LCD instead of b × d keeps numbers smaller.

FRACTION MULTIPLICATION & DIVISION

a/b × c/d = (a × c) / (b × d) and a/b ÷ c/d = a/b × d/c

Cancel common factors between any numerator and any denominator before multiplying to simplify computation.

SECTION 5

Detailed Breakdown — Converting Between Representations

Many TEAS problems present numbers in mixed formats — a whole number multiplied by a fraction, or a decimal subtracted from a mixed number. Converting between representations is therefore a prerequisite for correct computation. The diagram below summarizes the conversion pathways, and the accompanying table provides the procedural steps for each direction.

The three primary number formats — whole numbers, fractions, and decimals — plus mixed numbers, are connected by well-defined conversion procedures. Dashed green line indicates the fraction ↔ decimal conversion (divide numerator by denominator, or read the decimal as a fraction of a power of ten).

Conversion	Procedure	Example
Fraction → Decimal	Divide the numerator by the denominator.	3/8 → 3 ÷ 8 = 0.375
Decimal → Fraction	Write the decimal as a fraction over the appropriate power of 10, then simplify.	0.375 = 375/1000 = 3/8
Mixed Number → Improper Fraction	Multiply the whole number by the denominator, add the numerator, place over the original denominator.	2 3/4 → (2 × 4 + 3)/4 = 11/4
Improper Fraction → Mixed Number	Divide numerator by denominator; quotient is the whole part, remainder over denominator is the fractional part.	11/4 → 11 ÷ 4 = 2 R3 → 2 3/4
Whole Number → Fraction	Place the whole number over 1.	5 → 5/1

Conversion procedures between number formats

SECTION 6

Worked Example — Multi-Format Arithmetic

Consider a representative TEAS-style problem that combines all three number formats: Evaluate 3 1/2 + 2.75 − 4/5. This problem requires converting mixed numbers, decimals, and fractions into a common format, performing the operations, and simplifying the result.

Evaluate 3 1/2 + 2.75 − 4/5

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Step 1 — Convert all values to fractions

Convert 3 1/2 to an improper fraction: (3 × 2 + 1)/2 = 7/2. Convert 2.75 to a fraction: 2.75 = 275/100 = 11/4 (dividing numerator and denominator by 25). The third value is already a fraction: 4/5.

7/2 + 11/4 − 4/5
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Step 2 — Find the Least Common Denominator (LCD)

The denominators are 2, 4, and 5. The LCD is the least common multiple of these values. LCM(2, 4) = 4 and LCM(4, 5) = 20, so the LCD = 20.

LCD = 20
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Step 3 — Rewrite each fraction with denominator 20

7/2 = (7 × 10)/(2 × 10) = 70/20. 11/4 = (11 × 5)/(4 × 5) = 55/20. 4/5 = (4 × 4)/(5 × 4) = 16/20.

70/20 + 55/20 − 16/20
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Step 4 — Perform the addition and subtraction

Combine the numerators over the common denominator: (70 + 55 − 16)/20 = 109/20.

109/20
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Step 5 — Convert to a mixed number (if required) and verify

109 ÷ 20 = 5 remainder 9, so 109/20 = 5 9/20. As a decimal: 9 ÷ 20 = 0.45, so 5.45. Quick verification: 3.5 + 2.75 − 0.8 = 6.25 − 0.8 = 5.45. ✓

5 9/20 = 5.45

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Pro Tip

On the TEAS exam, converting everything to decimals can sometimes be faster than finding a common denominator, especially when the fractions have 'nice' decimal equivalents (halves, quarters, fifths, eighths). Always check the answer choices first — if they are in fractional form, stay in fractions to avoid rounding errors.

SECTION 7

Strengths & Pitfalls of Each Representation

Choosing between whole numbers, decimals, and fractions is not merely a matter of preference; each representation has distinct computational advantages and common pitfalls. The table below compares the three formats across several dimensions relevant to the TEAS exam.

Dimension	Whole Numbers	Decimals	Fractions
Addition / Subtraction	Straightforward columnar alignment; carry/borrow as needed.	Align decimal points; pad with zeros. Fast and intuitive.	Requires common denominator — an extra step that can introduce errors.
Multiplication	Standard algorithm; no place-value adjustment needed.	Multiply as whole numbers, then count total decimal places.	Multiply straight across — simplest algorithm of the three.
Division	Long division; may yield remainder.	Shift decimals to make divisor whole; may produce repeating decimals.	Invert and multiply — elegant but easy to forget to reciprocate.
Precision	Exact by nature.	May require rounding (e.g., 1/3 ≈ 0.3333).	Always exact — no rounding needed.
Common Pitfall	Forgetting to carry or borrow in multi-digit problems.	Misaligning the decimal point, especially in multiplication.	Forgetting to find a common denominator before adding/subtracting.

Comparison of computational characteristics across number formats

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KEY TAKEAWAY

Think of fractions, decimals, and whole numbers as three different 'languages' describing the same mathematical reality — like translating between English, French, and Spanish. Each has idioms that make certain tasks easier: fractions excel at exact representation (especially of thirds, sixths, and sevenths), decimals are ideal for quick estimation and comparison, and whole numbers anchor every calculation. Fluency in all three, and the ability to translate effortlessly, is the ultimate competitive advantage on the TEAS.

SECTION 8

Connection to Advanced Topics — Order of Operations & Algebraic Expressions

Arithmetic fluency is not an isolated skill; it is the engine that drives success in more complex TEAS topics such as the order of operations (PEMDAS), proportional reasoning, percentage calculations, and ultimately algebraic expression evaluation. When you encounter an expression like 2(3/4 + 1.5) − 0.25², every operation inside requires the same algorithms discussed in this lesson — but now sequenced according to a specific hierarchy. The table below contrasts basic arithmetic with these more advanced applications.

Feature	Basic Arithmetic	Advanced Applications
Operand Types	Whole numbers, decimals, fractions (standalone)	Variables, expressions, nested parentheses
Sequencing	Operations performed left to right, one at a time	PEMDAS dictates operation hierarchy
Error Source	Carrying, borrowing, decimal alignment, LCD	Incorrect operation order, sign errors in distribution
TEAS Relevance	Dosage calculations, measurement conversions	Ratio & proportion, percentage change, algebraic modeling

Basic arithmetic vs. advanced applications

As you progress through the Numbers and Algebra domain, you will find that every new topic — from solving linear equations to interpreting data sets — rests on the arithmetic foundation laid in this lesson. Investing in procedural fluency now yields compounding returns across the entire TEAS Mathematics section.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why you must find a common denominator before adding two fractions but not before multiplying them. Use the fractions 2/3 and 5/7 as an illustrative example.

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Addition combines 'like parts,' and fractions with different denominators represent parts of different-sized wholes. Converting 2/3 and 5/7 to the common denominator 21 (giving 14/21 and 15/21) ensures both fractions refer to the same unit partition, allowing the numerators to be summed directly: 14/21 + 15/21 = 29/21. Multiplication, however, operates on the concept of 'a fraction of a fraction' — 2/3 × 5/7 means '2/3 of 5/7,' which is computed by multiplying numerators (2 × 5 = 10) and denominators (3 × 7 = 21) independently, yielding 10/21. No alignment of unit size is needed because multiplication inherently creates a new partition.

Question 2: BASIC CALCULATION

Compute 14.36 + 7.8 − 9.125. Express your answer as a decimal.

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Align the decimal points and pad with zeros: 14.360 + 7.800 = 22.160. Then subtract: 22.160 − 9.125 = 13.035. The key step is ensuring all operands have the same number of decimal places (three in this case) to prevent misalignment errors. The answer is 13.035.

Question 3: INTERMEDIATE

Evaluate 5/6 ÷ 2/9 + 1.25. Express your answer as a mixed number in simplest form.

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First, perform the division: 5/6 ÷ 2/9 = 5/6 × 9/2 = 45/12. Simplify by dividing numerator and denominator by 3: 45/12 = 15/4. Next, convert 1.25 to a fraction: 1.25 = 5/4. Now add: 15/4 + 5/4 = 20/4 = 5. Alternatively, note that 15/4 = 3.75 and 3.75 + 1.25 = 5. The answer is 5, which can also be written as 5/1 or the mixed number 5 0/1. The problem tests the order of operations (division before addition) and conversion between decimals and fractions.

Question 4: APPLIED

A nurse administers 2 1/2 tablets of medication three times daily. Each tablet contains 0.125 grams of the active ingredient. How many total grams of the active ingredient does the patient receive per day? Express your answer as a decimal.

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The patient receives 2 1/2 tablets per dose. Convert to a decimal or improper fraction: 2 1/2 = 2.5 tablets. Over three doses: 2.5 × 3 = 7.5 tablets per day. Each tablet contains 0.125 g, so the total daily intake is 7.5 × 0.125 g. Compute: 7.5 × 0.125 = 7.5 × 125/1000 = 937.5/1000 = 0.9375 g. Alternatively, 0.125 × 7 = 0.875 and 0.125 × 0.5 = 0.0625, so 0.875 + 0.0625 = 0.9375 g. The patient receives 0.9375 grams of the active ingredient daily.

Question 5: CRITICAL THINKING

A student claims that 0.1 + 0.1 + 0.1 must equal exactly 0.3 because 1/10 + 1/10 + 1/10 = 3/10. While this is arithmetically true, explain why a computer might report a result slightly different from 0.3 for this sum, and discuss the implications for precision-sensitive fields such as pharmacology. What strategy can a test-taker use to avoid analogous rounding pitfalls on the TEAS?

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In the decimal system, 0.1 is exact. However, computers store numbers in binary (base 2), and 1/10 has an infinitely repeating binary expansion (0.0001100110011…), much as 1/3 = 0.3333… in base 10. Repeated addition of an approximate 0.1 in binary accumulates rounding error, so the computer sum may be 0.30000000000000004 rather than 0.3 exactly. In pharmacology, such floating-point imprecision is typically mitigated by using integer arithmetic in the smallest unit (e.g., micrograms instead of grams) or by using exact rational arithmetic. For the TEAS, the analogous pitfall is working with approximate decimal representations of fractions like 1/3 or 1/6. The safest strategy is to stay in fractional form throughout the computation and convert to a decimal only at the final step, thereby preserving exactness and avoiding accumulated rounding errors.

SUMMARY

Lesson Summary

Summary

This lesson established complete fluency in the four fundamental arithmetic operations — addition, subtraction, multiplication, and division — across whole numbers, decimals, and fractions. Key principles include the commutative, associative, and distributive properties, which govern how operations may be reordered and regrouped for efficiency.

For decimal operations, always align decimal points in addition and subtraction, and count total decimal places in multiplication. For fraction operations, find the least common denominator before adding or subtracting, and use the invert-and-multiply strategy for division. When problems mix formats, convert all values to a single representation — typically fractions for exact work or decimals for speed — before computing. These skills form the non-negotiable foundation for every quantitative task on the TEAS examination, from dosage calculations to algebraic reasoning.