Study guide: Fractional Rates and Proportional Relationships
Proportional relationships compare multiplicative changes rather than fixed additions. Rates connect two labeled quantities, and a proportional graph must be consistent with a zero starting amount as well as a constant ratio.
Ideas to explain in your own words
- A unit rate asks how much of one quantity corresponds to one whole unit of another. Dividing fractions requires multiplying by the reciprocal of the divisor. Keep distance over time when finding speed.
- A proportional rule has the form y=kx. Every nonzero input must give the same output-to-input ratio; an isolated matching pair cannot establish the whole table’s structure. Check every supplied nonzero row, and check that a stated row with input zero has output zero.
- The point (1,k) on a proportional graph describes its constant. Another nonzero point gives the same k by dividing its vertical coordinate by its horizontal coordinate. A rising line is not automatically proportional.
- Rates can be compared fairly after both are expressed per one identical unit. A larger total can still come from a slower process if it had more time. Use the same time unit and the same output unit for both processes.
Useful vocabulary
- unit rate
- An amount of one quantity per one unit of another.
- constant of proportionality
- The fixed k in y=kx.
- origin
- The coordinate point (0,0).
Watch your reasoning
- Keep distance over time when finding speed. Switching the order produces hours per kilometer, which answers a different question; label the result so its meaning remains visible.
- Check every supplied nonzero row, and check that a stated row with input zero has output zero. Constant differences alone do not distinguish a proportional line from a line with an intercept.
- A rising line is not automatically proportional. The stated graph must pass through the origin, and the roles of input and output must stay consistent throughout the calculation.
A worked review example
Complete supplied guide examples: Divide Fractional Distances by Fractional Times — A model cart travels 1 kilometers in 3/4 hours at constant speed. What is its unit rate? Candidate answers: (A) 3.3333 km/hour (B) 3/4 km/hour (C) 4/3 km/hour (D) 7/4 km/hour. Divide distance by time: (1)÷(3/4)=4/3. The denominator is hours, so the rate is kilometers per hour. Test a Whole Table for Proportionality — For candidate multiplier 2, which supplied (x,y) table is proportional? Candidate answers: (A) (0,1), (1,2), (2,4) (B) (1,2), (2,2), (4,2) (C) (1,2), (2,4), (4,8) (D) (1,2), (2,5), (4,8). The correct table has y/x=2 for every nonzero x. The other tables either change this ratio or imply a nonzero value at x=0. Read a Proportional Graph’s Constant — A proportional graph passes through (0,0) and (2,6). What is y when x=4? Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24. The constant of proportionality is y/x=6/2=3. Use y=3x, so y=3×4=12. Compare Rates with Different Totals — Printer A makes 6 pages in 2 minutes; B makes 16 in 5 minutes. Which statement follows from these constant-rate models? Candidate answers: (A) A is faster because its time is smaller (B) B is faster: 16/5 versus 3 pages/minute (C) The page totals alone determine the faster rate (D) The rates are equal. A has 6/2=3 pages/minute. B has 16/5=16/5, which exceeds 3. Comparing equally labeled unit rates supports B as faster.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.