Skip to main content

Repair: Arithmetic, Fractions and Units

Totals and averages​

A total adds measurements with the same unit. For sessions of 10, 20 and 30 minutes, the total is 10 + 20 + 30 = 60 minutes. The arithmetic mean is total divided by count: 60 minutes / 3 = 20 minutes per session. A mean need not be an actual observed value. For an empty collection, the total can be zero, but the mean is undefined because division by zero is not allowed. Do not silently treat an empty mean as zero.

Fractions, decimals and percentages​

A fraction compares part with whole. Finishing 2 of 8 exercises gives 2/8 = 1/4 = 0.25 = 25%. Simplify by dividing numerator and denominator by the same nonzero number. A percentage means “per hundred,” so multiply a decimal proportion by 100 to express it as a percentage.

For addition, match denominators: 1/2 + 1/4 = 2/4 + 1/4 = 3/4. Adding denominators would incorrectly give 2/6. Check whether the result is plausible: one-half plus something positive must be greater than one-half.

Units as a check​

Use 60 minutes / 1 hour as a conversion factor. Thus 2.5 hours × 60 minutes/hour = 150 minutes; hours cancel. Likewise 1,800 bytes / (300 bytes/record) = 6 records. Adding hours to minutes without converting first is a units error, not just an arithmetic slip. A quantity can have the right digits and the wrong unit.

Order, signs and powers​

Parentheses first, then powers, then multiplication/division, then addition/subtraction. Within the same level, evaluate left to right. 6 + 3 × 2 = 12, while (6 + 3) × 2 = 18. In mathematics 2^4 means four factors of 2, giving 16; 2 × 4 gives 8. In Python exponentiation uses **; ^ has a different meaning.

A negative value represents direction or signed difference. If a balance changes from 5 to 2, the change is 2 - 5 = -3; adding 3 would reverse the direction.

Practice before checking​

Calculate the total and mean of 12, 18 and 30 minutes; express 6/15 as a simplified fraction, decimal and percentage; convert 0.75 hours to minutes; divide 3,600 bytes into 400-byte records; compare 4 + 3 × 2 with (4 + 3) × 2; calculate 2^6.

Expected results: 60 minutes and 20 minutes/session; 2/5, 0.4, 40%; 45 minutes; 9 records; 10 versus 14; 64. Explain a mismatch before retrying. Use a calculator to check a prediction rather than replace the reasoning.

Complete domain B of the fresh reassessment. For more examples, use the relevant fractions, decimals or measurement section of OpenStax Prealgebra 2e; reading the whole book is optional and outside this repair estimate.