1.1 Arithmetic, Fractions, Decimals, and Percentages

Introduction to Arithmetic Reasoning

Arithmetic reasoning forms the backbone of the ASTB-E Math Skills Test. Because you are not permitted to use a calculator, you must be exceptionally comfortable with mental math and manual calculations. The foundation of all arithmetic is the Order of Operations, universally remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Mastery of this hierarchy ensures that complex expressions are evaluated correctly. For instance, in the expression 8 + 4 × (3 - 1)^2, you first evaluate the parentheses (3 - 1) = 2, then apply the exponent 2^2 = 4, perform multiplication 4 × 4 = 16, and finally add 8 + 16 = 24. Without strict adherence to PEMDAS, the result would be wildly incorrect.

Fractions: The Language of Parts

Fractions represent a part of a whole and consist of a numerator (the top number) and a denominator (the bottom number). Adding and subtracting fractions requires a common denominator. To add 1/4 and 2/3, you must find the least common multiple of 4 and 3, which is 12. Thus, 1/4 becomes 3/12 and 2/3 becomes 8/12. The sum is 11/12. Multiplying fractions is more straightforward: simply multiply the numerators together and the denominators together. For example, 3/4 × 2/5 = 6/20, which simplifies to 3/10. Dividing fractions requires the "keep-change-flip" method: keep the first fraction, change the division sign to multiplication, and flip the second fraction (find its reciprocal). Dividing 3/4 by 2/5 becomes 3/4 × 5/2 = 15/8, an improper fraction that can be expressed as the mixed number 1 7/8.

Complex fractions—fractions where the numerator, denominator, or both contain a fraction—often appear intimidating but follow the same rules as basic division. To simplify a complex fraction like (1/2) / (3/4), you apply the keep-change-flip rule to get 1/2 × 4/3 = 4/6, which simplifies to 2/3. When adding or subtracting mixed numbers, it is generally safer to convert them to improper fractions first. This eliminates the confusion of borrowing from whole numbers in subtraction.

Decimals and Conversions

Decimals are simply fractions with denominators that are powers of 10. Adding and subtracting decimals requires aligning the decimal points. When multiplying decimals, ignore the decimal points initially, multiply the numbers as if they were whole, and then count the total number of decimal places in the original factors to place the decimal point in the product. For 0.4 × 0.12, multiply 4 by 12 to get 48, then account for three total decimal places to yield 0.048. For division, shift the decimal point in the divisor to the right to make it a whole number, and shift the decimal in the dividend by the same number of places. Conversions between fractions and decimals are frequent on the ASTB-E. You should memorize common conversions:

Percentages in Practice

Percentages represent parts per hundred. To convert a decimal to a percentage, multiply by 100 (shift the decimal two places to the right) and add the percent sign. Conversely, to convert a percentage to a decimal, divide by 100. Calculating a percentage of a number is done by converting the percent to a decimal or fraction and multiplying. For example, to find 15% of 80, you can calculate 0.15 × 80 = 12. Alternatively, use the 10% rule: 10% of 80 is 8, and 5% is half of that (4); adding them gives 12. Percentage increase and decrease are crucial for word problems. The formula is (New - Old) / Old × 100. If an item’s price drops from $50 to $40, the decrease is $10. The percentage decrease is 10 / 50 = 0.20, or 20%.

Ratios and Proportions

A ratio is a comparison of two quantities, often expressed as a fraction or with a colon (e.g., 3:4). A proportion is an equation stating that two ratios are equal. Solving proportions generally involves cross-multiplication. For instance, if a recipe requires 2 cups of sugar for every 3 cups of flour, and you are using 9 cups of flour, the proportion is 2/3 = x/9. Cross-multiplying yields 3x = 18, so x = 6. Ratios are frequently tested in word problems involving speed, work rates, and mixtures. The key is to ensure consistent units across the numerator and denominator on both sides of the proportion.

Word Problems and Mental Math Strategies

Word problems test your ability to translate English into mathematical equations. When approaching a word problem, first identify the unknown and assign it a variable. Look for keywords: "is" means equals, "of" signifies multiplication, "more than" implies addition, and "per" denotes division.

Distance, rate, and time (D=RT) problems are a staple of the ASTB-E. If an aircraft travels at 400 knots for 2.5 hours, the distance is 400 × 2.5 = 1000 nautical miles. Work-rate problems are another common variation. If pump A can fill a tank in 3 hours, its rate is 1/3 of the tank per hour. If pump B can fill the tank in 6 hours, its rate is 1/6. Working together, their combined rate is 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2. Thus, together they can fill the tank in 2 hours.

Because calculators are prohibited, look for ways to simplify calculations. Use estimation to eliminate answer choices that are logically out of bounds. For example, if you are multiplying 48 by 21, you can estimate this as 50 by 20, which is 1000. Any answer choice far from 1000 can be quickly eliminated. Additionally, practice decomposing numbers: 48 × 21 is 48 × (20 + 1) = 960 + 48 = 1008. Building these mental math muscles will save you precious time during the test.