OSAT Middle Level Mathematics (225)

The base-ten positional system assigns value based on digit position. In the number 4,725.38, the digit 4 represents 4 × 1000, the digit 7 represents 7 × 100, the digit 2 represents 2 × 10, the digit 5 represents 5 × 1, the digit 3 represents 3 × 0.1, and the digit 8 represents 8 × 0.01.

Each position represents a power of 10. Moving left increases by powers of 10; moving right decreases by powers of 10. Understanding place value supports operations with multi-digit numbers and decimals.

Converting between bases requires understanding positional notation. To convert from base ten to another base, repeatedly divide by the new base and record remainders.

A fraction represents a part of a whole or a ratio of two quantities. In the fraction a/b, a is the numerator (parts taken) and b is the denominator (total equal parts).

Decimals extend place value to represent parts of a whole using negative powers of 10. The decimal 0.375 means 3 tenths + 7 hundredths + 5 thousandths.

Converting Fractions to Decimals: Divide numerator by denominator. Results are either terminating (1/4 = 0.25) or repeating (1/3 = 0.333...).

Converting Decimals to Fractions: For terminating decimals, use place value (0.75 = 75/100 = 3/4). For repeating decimals, use algebraic methods.

Percent means "per hundred." To convert percent to decimal, divide by 100 (75% = 0.75). To convert decimal to percent, multiply by 100 (0.42 = 42%). To convert percent to fraction, place over 100 and simplify (25% = 25/100 = 1/4).

Percent Problems: Three main types exist. Finding the percent: What percent of 80 is 20? (20/80 = 0.25 = 25%). Finding the part: What is 30% of 150? (0.30 × 150 = 45). Finding the whole: 24 is 40% of what number? (24/0.40 = 60).

Percent Change: Calculate as (new value - original value)/original value × 100. Positive result indicates increase; negative indicates decrease.

Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of 10. Standard form: a × 10^n where 1 ≤ a < 10 and n is an integer.

Large numbers have positive exponents: 4,500,000 = 4.5 × 10^6. Small numbers have negative exponents: 0.00032 = 3.2 × 10^-4.

Operations in Scientific Notation: To multiply, multiply coefficients and add exponents. To divide, divide coefficients and subtract exponents. Adjust the result if the coefficient falls outside the range [1, 10).