GED · Mathematical Reasoning · 5 min read · Open lesson
Whole Numbers, Fractions, Decimals, and Units
- Apply operations to signed numbers and evaluate multi-step mathematical expressions using the standard order of operations.
- Perform fraction and mixed number conversions, arithmetic, and simplifications by hand and with calculator tools.
- Solve decimal arithmetic, consumer transactions, and unit conversions across US customary and metric systems.
Signed Numbers, Absolute Value, and Order of Operations
A clear understanding of real numbers begins along the number line. A negative number represents a quantity less than zero on a number line. In contrast, the absolute value of a number represents its distance from zero and is never negative. Because distance cannot be negative, taking the absolute value of any positive or negative value yields a non-negative measurement.
When performing arithmetic with positive and negative numbers, consistent rules determine the final sign. Adding two numbers with the same sign produces a sum with that common sign. To add numbers with different signs, subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value; if the absolute values are equal, the sum is zero. Subtracting an integer is equivalent to adding its opposite. For multiplication and division, the product or quotient of two numbers with identical signs is always positive, whereas the product or quotient of two numbers with opposite signs is always negative. Furthermore, multiplying any real number by zero results in zero, while dividing any number by zero is mathematically undefined.
Evaluating multi-step calculations requires following the precise order of operations. According to the order of operations, expressions inside grouping symbols are evaluated first. Exponents are evaluated before multiplication, division, addition, or subtraction. Multiplication and division carry equal priority and are evaluated from left to right. Finally, addition and subtraction carry equal priority and are evaluated from left to right.
Fractions, Mixed Numbers, and Operations
A fraction can represent part of a whole or a division; its numerator is above the fraction bar, and its nonzero denominator is below. Any whole number can be converted to a fraction by placing it over a denominator of 1. Positive fractions fall into distinct categories based on the relationship between their terms. For positive fractions, a proper fraction has a numerator smaller than its denominator. Conversely, for positive fractions, an improper fraction has a numerator greater than or equal to its denominator. A mixed number combines a non-zero whole number with a proper fraction. To convert a positive mixed number to an improper fraction, multiply the whole-number part by the denominator, add the numerator, and place the result over the original denominator.
Adding and subtracting fractions requires careful attention to the denominator. Fractions must share a common denominator before they can be added or subtracted. The least common denominator is the smallest positive multiple shared by all denominators in the set. When adding or subtracting fractions with common denominators, combine the numerators and keep the denominator unchanged. The denominator remains untouched throughout the addition or subtraction step.
Multiplying and dividing fractions follow direct arithmetic procedures. Multiplying fractions involves multiplying the numerators together and the denominators together. Division utilizes reciprocals. The reciprocal of a non-zero fraction is obtained by swapping its numerator and denominator. To divide one fraction by another, multiply the dividend by the reciprocal of the divisor. Once any operation is complete, a fraction is in simplest form when the greatest common factor of the numerator and denominator is 1.
Decimals and Consumer Financial Applications
Decimals represent parts of a whole through place values established to the right of the decimal point. In a decimal number, the position immediately to the right of the decimal point is the tenths place, and the second position to the right of the decimal point is the hundredths place. Trailing zeros added to the right of the last non-zero digit after a decimal point do not change the number's numerical value. In everyday financial contexts, amounts in US dollars are usually expressed to two decimal places, with the hundredths place representing cents.
Manual arithmetic with decimals demands exact alignment and digit counting. Adding or subtracting decimals by hand requires aligning the numbers vertically by their decimal points. When multiplying decimals by hand, add the numbers of decimal places in the factors to place the decimal point in the product; trailing zeros in the result can then be removed. To divide by a nonzero decimal using long division, move the decimal point the same number of places in both the divisor and dividend until the divisor is a whole number.
Practical consumer mathematics relies heavily on decimal operations. Unit price is calculated by dividing the total price of an item by the quantity or weight purchased. When purchasing goods, a discount amount is found by multiplying the original retail price by the discount percentage as a decimal, and the sale price equals the original price minus the discount amount. Sales tax is calculated by multiplying the subtotal by the sales tax rate written as a decimal. If sales tax is the only additional charge, the total purchase price equals the subtotal plus the sales tax. Finally, when the amount paid covers the cost, change equals the amount paid minus the total purchase cost.
Measurement Conversions and Dimensional Analysis
Measurement systems quantify real-world quantities using established standard units. In the US customary system, length relies on linear units where 1 foot is equal to 12 inches, 1 yard is equal to 3 feet or 36 inches, and 1 mile is equal to 5,280 feet. Customary weight establishes that 1 pound is equal to 16 ounces, and 1 ton is equal to 2,000 pounds. Customary capacity measures volume progressively: 1 cup contains 8 fluid ounces, 1 pint equals 2 cups, 1 quart equals 2 pints, and 1 gallon is equal to 4 quarts.
The metric system is a base-10 measurement system using prefixes to indicate powers of ten. In the metric system, linear measures scale uniformly: 1 meter is equal to 100 centimeters, 1 meter is equal to 1,000 millimeters, and 1 kilometer is equal to 1,000 meters. Metric mass and capacity follow similar powers of ten: 1 kilogram is equal to 1,000 grams, and 1 liter is equal to 1,000 milliliters.
Converting between units requires understanding whether the target unit is larger or smaller. Converting from a larger unit of measurement to a smaller unit requires multiplication by the conversion factor. Converting from a smaller unit of measurement to a larger unit requires division by the conversion factor. Alternatively, dimensional analysis converts units by multiplying by fractions that equal one to cancel unwanted units.
| Measurement System | Unit Equivalence | Relationship |
|---|---|---|
| US Customary Length | 1 foot | 12 inches |
| US Customary Length | 1 yard | 3 feet or 36 inches |
| US Customary Length | 1 mile | 5,280 feet |
| US Customary Weight | 1 pound | 16 ounces |
| US Customary Weight | 1 ton | 2,000 pounds |
| US Customary Capacity | 1 cup | 8 fluid ounces |
| US Customary Capacity | 1 pint | 2 cups |
| US Customary Capacity | 1 quart | 2 pints |
| US Customary Capacity | 1 gallon | 4 quarts |
| Metric Length | 1 meter | 100 centimeters |
| Metric Length | 1 meter | 1,000 millimeters |
| Metric Length | 1 kilometer | 1,000 meters |
| Metric Mass | 1 kilogram | 1,000 grams |
| Metric Volume | 1 liter | 1,000 milliliters |
Common mistakes
- Using the subtraction operation key instead of the separate negative key on the TI-30XS calculator, which triggers a syntax error.
- Attempting to add or subtract fractions without first establishing a common denominator.
- Dividing by zero instead of recognizing that dividing any number by zero is mathematically undefined.
- Multiplying rather than dividing when converting a smaller measurement unit to a larger measurement unit.
Summary
- Absolute value is the distance from zero and is never negative, while negative numbers represent quantities less than zero.
- The order of operations prioritizes grouping symbols first, followed by exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.
- Adding and subtracting fractions requires a common denominator, while multiplying fractions involves multiplying numerators together and denominators together, and dividing fractions uses multiplication by the reciprocal.
- A fraction is in simplest form when the greatest common factor of the numerator and denominator is 1, a display feature supported on the TI-30XS MultiView.
- In decimals, the tenths place is immediately right of the decimal point, the hundredths place is second, and adding trailing zeros to the right does not change the numerical value.
- Consumer math uses decimals to find unit prices, sales tax, total purchase prices, discounts, sale prices, and change.
- Unit conversions require multiplying when changing from larger to smaller units and dividing when changing from smaller to larger units, which can be accomplished via dimensional analysis or values on the formula sheet.
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