Ratio and Proportions

Proportions, Ratios, and Unit Rates: Key Concepts from the Transcript

• Context and purpose

  • Proportions relate two ratios that describe a relationship between two quantities.

  • The order of the terms matters: in a ratio, the first quantity corresponds to the first term, the second quantity to the second term.

  • Common goal: solve for a missing quantity in a proportional relationship, or convert a ratio to a unit rate.

• Core ideas to remember

  • Order matters: if a ratio says Sugar to Flour is 2 to 5, then 2 corresponds to sugar and 5 corresponds to flour.

  • Write ratios as fractions to compare or solve: if the ratio is 2:5, you can write it as $\frac{2}{5}$.

  • Always keep track of what each number represents; labeling helps avoid confusion when solving.

  • When solving proportions, you must include an equals sign between the two fractions; do not write something like $\frac{2}{5} = 0$.

• Proportions and cross-multiplication

  • Set up a proportion with the known ratio on one side and the unknown on the other:

  • Example setup: $\frac{2}{5} = \frac{7}{x}$ where 2 = sugar, 5 = flour, 7 = sugar amount, and x = flour amount.

  • Cross-multiply to solve: the product of the numerator of one side and the denominator of the other side equals the product of the denominator of the first side and the numerator of the second side:

  • $2x = 5\cdot 7 = 35$

  • Solve for the unknown by dividing both sides by the coefficient of the unknown:

  • $x = \frac{35}{2} = 17.5$

  • Key caution: always include the equal sign in the equation when cross-multiplying; a common mistake is to omit it or to set up as a non-equal expression.

• Worked example 1: Cookies (sugar to flour)

  • Scenario: A cookie recipe uses a ratio of 2 parts sugar to 5 parts flour. If you have 7 cups of sugar, how much flour is needed?

  • Setup: $\frac{2}{5} = \frac{7}{x}$

  • Cross-multiply: $2x = 5\cdot 7 = 35$

  • Solve: $x = \frac{35}{2} = 17.5\text{ cups of flour}$

  • Note on units: keep labeling explicit (sugar = 7 cups; flour = x cups) so you don’t confuse the quantities.

• Worked example 2: Driving distance with fuel (miles per gallon)

  • Scenario: If you used 18 gallons to go 450 miles, how far can you go with 6 gallons?

  • Ratio setup: $\frac{18}{450} = \frac{6}{x}$ where x is miles per 6 gallons.

  • Cross-multiply: $18x = 6\cdot 450 = 2700$

  • Solve: $x = \frac{2700}{18} = 150$

  • Conclusion: With 6 gallons, you can travel 150 miles.

  • Important note: ensure the equation has the equal sign; rearranging without the equal sign leads to incorrect results.

• Worked example 3: Medicine dosing (scale with weight)

  • Scenario: If the recommended dose is 400 mg for 150 pounds, what is the dose for 275 pounds?

  • Set up proportional relationship: $\frac{400}{150} = \frac{x}{275}$

  • Cross-multiply: $150x = 400\cdot 275 = 110{,}000$

  • Solve: $x = \frac{110{,}000}{150} = 733.333\overline{3}\text{ mg}$

  • Significance: demonstrates dose scaling by body weight; always verify units and whether rounding is appropriate in context.

• Building ratios from a budget (data table)

  • Scenario: Create simplified ratios from a monthly budget.

  • Given example categories and amounts (hypothetical):

  • Food: $300, Transportation: $200, Supplies: $50, Phone: $100, Clothes: $75, Car (insurance+gas): $200, Leisure: $150.

  • Forming a ratio (food to transportation):

  • Numerator corresponds to the first category (food), denominator to the second (transportation): $$\frac{300