Fractions, decimals and arithmetic operations

01 / UNDERSTAND & APPLY

A fraction is a division with a nonzero denominator

a/b means a divided by b, with b≠0. Multiplying numerator and denominator by the same nonzero number preserves the value. Cancelling is division by a common factor of the entire numerator and denominator; terms separated by addition cannot be cancelled independently.

A negative sign can sit in the numerator, denominator or in front of the fraction. Two negative signs cancel. An improper fraction is a legitimate exact answer, not an error. Mixed numbers are sums: −2⅓ means −(2+1/3)=−7/3.

02 / UNDERSTAND & APPLY

Common denominators create comparable parts

To add or subtract fractions, express them in the same units by choosing a common denominator, ideally the least common multiple. Add numerators only after rewriting both fractions; the shared denominator remains. Reduce at the end or earlier if convenient.

A result should have a sensible sign and magnitude. Adding two positive fractions must exceed each one separately; subtracting a larger positive fraction from a smaller one must give a negative result.

03 / UNDERSTAND & APPLY

Cancel factors and invert only the divisor

Multiply numerators together and denominators together. Before multiplying, cancel common factors across numerator and denominator to avoid large arithmetic. Division by a nonzero fraction multiplies by its reciprocal, because the reciprocal product equals 1. The fraction being divided is not inverted.

Division by a number between 0 and 1 increases a positive quantity; multiplication by such a number decreases it. A large-looking denominator is not enough to compare fractions unless their numerators are equal and positive.

04 / UNDERSTAND & APPLY

Place value is an exact power-of-ten fraction

A terminating decimal is a fraction over a power of 10: 0.0375=375/10000=3/80. Align decimal points for addition and subtraction. For multiplication, multiply integer digits and then restore the combined number of decimal places; estimate first to detect a misplaced point.

To divide by a decimal, multiply dividend and divisor by the same power of 10 until the divisor is an integer. A calculator display can round repeating values; a displayed 0.333333 does not prove an exact value of 333333/1000000.

05 / UNDERSTAND & APPLY

The reduced denominator predicts termination

A rational number has a terminating or eventually repeating decimal expansion. After a fraction is reduced, its decimal terminates precisely when the positive denominator has no prime factors other than 2 and 5. Those factors can be completed into a power of 10. A denominator containing 3 or 7 forces repetition unless the factor first cancels.

To convert a repeating decimal, shift one full repeating block and subtract to eliminate the infinite tail. Leading nonrepeating digits require two shifts. Do not round a repeating decimal before using it in an equality.

06 / UNDERSTAND & APPLY

Grouping determines which operation occurs

Evaluate parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right. Multiplication does not automatically precede division. A fraction bar groups the whole numerator and whole denominator. Use parentheses to reproduce that grouping on a calculator.

−3² means −(3²)=−9, whereas (−3)²=9. A minus sign before parentheses distributes across every term. When numerical signs seem confusing, determine the sign of a product first, then calculate its magnitude.

07 / UNDERSTAND & APPLY

Compare exactly, estimate strategically

For positive denominators, a/b<c/d exactly when ad<bc. If a denominator is negative, first move its sign to the numerator. Useful benchmarks include 0, 1/2 and 1. If two positive fractions lie on opposite sides of 1/2, a full decimal calculation is unnecessary.

Rounding is useful for rejecting implausible choices but can erase a small genuine difference. Two quantities that both round to 0.67 need not be equal. For QC, preserve an exact difference or cross-product when the quantities are close.

08 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

09 / UNDERSTAND & APPLY

Original practice with exact answers

Do not round unless requested.