Operations with Integers, Ratios and Rates

Integers are whole numbers and their opposites, including zero. Positive integers are greater than zero (e.g., 1, 2, 3, ...), and negative integers are less than zero (e.g., -1, -2, -3, ...). Zero is neither positive nor negative.

When adding integers with the same sign, add their absolute values and keep the common sign. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Subtracting an integer is the same as adding its opposite. For example, a - b is equivalent to a + (-b), and a - (-b) is equivalent to a + b.

When multiplying or dividing integers: if the signs are the same (both positive or both negative), the result is positive. If the signs are different (one positive and one negative), the result is negative.

A ratio is a comparison of two or more quantities of the same type. Ratios do not have units. They can be written using a colon (e.g., a : b) or as a fraction (e.g., a/b).

To simplify a ratio, divide all parts of the ratio by their greatest common divisor (GCD) until the parts are whole numbers with no common factors other than 1. This is similar to simplifying fractions.

Equivalent ratios represent the same comparison. You can find equivalent ratios by multiplying or dividing all parts of the ratio by the same non-zero number.

A rate is a comparison of two quantities of different types. Unlike ratios, rates have units (e.g., kilometres per hour, dollars per kilogram).

A unit rate is a rate where the second quantity is expressed as a single unit. For example, '80 kilometres per hour' is a unit rate because it specifies the distance travelled in 1 hour.