Ratio and Proportion Concepts

Ratio and Proportion Notes

Ratios and Rates

Ratios

• Definition of Ratios: A ratio is the comparison of two quantities that have the same units.

  • Example: 250 ml to 500 ml can be expressed as follows:

    • As a phrase: 250 to 500

    • As a fraction: $\frac{250}{500}$

    • In colon format: 250:500

Simplest Form of a Ratio

• Definition: A ratio is in simplest form when the two numbers do not have a common factor and both numbers are whole numbers.

• Example 1: 250:500 can be simplified to 1:2.

  • Detailed step: $\frac{250}{500} = \frac{1}{2}$

• Example 2: The ratio 35:10 simplifies to 7:2.

  • Detailed step: $\frac{35}{10} = \frac{7}{2}$

  • Question: Write 35:10 in simplest form.

• Example 3: The ratio 2800:400 simplifies through division by the common factor (400).

  • Detailed step: $\frac{2800}{400} = 7:1$

Specific Applications of Ratios

Example of Tax Withholding

• Scenario: Lina earns $500 weekly. Out of that gross pay:

  • $125 is withheld for federal taxes

  • $60 is withheld for provincial taxes

• Calculations:

  • Provincial Taxes to Gross Pay Ratio:

    • Ratio = $\frac{60}{500} = \frac{3}{25}$

    • Simplified ratio: provincial taxes : gross pay = 3 : 25

  • Federal Taxes to Gross Pay Ratio:

    • Ratio = $\frac{125}{500} = \frac{1}{4}$

    • Simplified ratio: federal taxes : gross pay = 1 : 4

Rates

Definition of Rates

• Definition: A rate is a comparison of two quantities that have different units.

• Example: 18 ounces to $1.50 results in a unit rate of $\frac{1.50}{18} = 0.25$ dollars per ounce.

Unit Rates

• Definition: A unit rate is the rate for a single unit.

• Example: From the previous rate, 0.25 dollars is the unit rate for 1 ounce.

• Notation as Fraction: Convert rates to lowest terms. For instance, for 6 minutes at $32, the rate would be calculated as $\frac{32}{6} = \frac{16}{3}$.

The Concept of Proportions

Proportions

• Definition: A proportion states that two ratios or rates are equal.

• Example: The proportion 16/12 = 4/3 can be expressed as “sixteen is to twelve as four is to three.”

• Notation: Write the proportion of 5.6 to 4.4 as 112 to 88.

Equality Test for Proportions

• Method: The equality test for proportions is used to determine if a statement is a proportion via cross-products.

  • Example: $1 \cdot 8 = 8 \cdot 1$ results in equal products indicating these are proportional.

• Second Example: 75 km in 5 hours compared to 105 km in 7 hours:

  • First cross product: $75 \cdot 7 = 525$

  • Second cross product: $105 \cdot 5 =525$

  • Conclusion: These two rates are equal, thus it forms a proportion.

Solving Proportions

General Approach

• Definition of a Variable: A variable is represented by a letter that stands for a number we do not yet know.

• To solve for n when given proportions in the form $a \cdot n = b$, divide each side by the coefficient of n.

Example Problems

• Example 1: Solve for n in $8 \cdot n = 72$

  • Rearranging leads to: $n = \frac{72}{8} = 9$.

• Example 2: Solve for n in $n \cdot 11.4 = 57$

  • Solution: $n = \frac{57}{11.4} = 5$. Verify by multiplying back to check: $5 \cdot 11.4 = 57.$

• Example 3: Finding missing numbers can be solved by finding cross products such as in the ratio 15/4 = 6/n.

  • Completing step by cross-multiplying, it leads to $4 * 6 = 15*n$ which can be solved afterwards.

Solving Applied Problems Involving Proportions

Steps for Problem Solving

1. Understand the Problem:

   • Actions: Read carefully, visualize with a drawing if needed, and establish a math blueprint.

2. Solve and State the Answer:

   • Actions: Perform calculations clearly and present the answer with appropriate units.

3. Check:

   • Actions: Estimate the answer logically and compare it with the computed result to verify its validity.

Examples in Context

• Example 1: A baseball pitcher gave up 52 earned runs in 260 innings: to calculate runs in 9 innings, apply proportional reasoning.

• Example 2: Recommended ratios such as 2 gallons for 750 square feet: Calculate the cost based on a painter’s requirement to paint 7875 square feet, noting paint costs as well.

Understanding Percent

Definition of Percent

• Meaning: The term percent means “per hundred.”

• Explanation: A percent is a comparative measure of a part of a complete whole. The statement “7 of 100 rectangles are shaded” conveys this as 7%.

Conversion Between Formats

• Example Enacting Denominators: To write $\frac{59.6}{100}$, convert to the corresponding percent and simplify for clarity.

Changing Between Percents, Decimals, and Fractions

• Percent to Decimal Conversion: Drop the % and shift the decimal point left by two places.

  • Conversion Example: The percent 27% converts to 0.27.

• Decimal to Percent Conversion: Shift the decimal point two places right and add the % symbol.

  • Conversion Example: The decimal 0.25 converts to 25%.

• Fraction to Percent: First convert the fraction to a decimal, then from decimal to percent.

  • Example: $\frac{7}{8} = 0.875$ converts to 87.5%.

• Equivalent Forms: Understanding the conversions among fractions, decimals, and percents is crucial to mastering proportions. Adjusting notation provides multimedia skills in the mathematics of comparisons.

Summary of Conversions

• Create a comparative table identifying equivalent notations for fractions, decimals, and percents.