7th Grade Mathematics — Pre-Algebra — Foundations of Mathematical Thinking
Scaling Relationships and Understanding Parts of the Whole
A proportion is an equation that states two ratios are equal. For example, 2/3 = 8/12 is a proportion because both ratios simplify to the same value. Proportions are powerful tools for solving problems where two quantities maintain a constant ratio as they change in size.
Proportional thinking is everywhere: enlarging a photograph without distortion, calculating medicine dosages based on body weight, converting currencies, or scaling a recipe. Whenever two quantities change at the same rate, they are in proportion.
To solve a proportion with an unknown, we use cross multiplication. In the proportion a/b = c/d, the cross products are equal: ad = bc. For example, to solve x/6 = 5/15, cross multiply: 15x = 30, so x = 2.
Cross multiplication works because multiplying both sides of an equation by the same quantities preserves equality. It transforms a proportion into a simple one-step equation that we already know how to solve. Always check your solution by substituting it back and confirming that both ratios are indeed equal.
Percent means 'per hundred.' It is a ratio that compares a number to 100. So 25% means 25 out of 100, or 25/100, which simplifies to 1/4. Percents, fractions, and decimals are three different ways to express the same value: 50% = 1/2 = 0.5.
To convert a fraction to a percent, divide the numerator by the denominator and multiply by 100. To convert a percent to a decimal, divide by 100 (move the decimal point two places left). These conversions are essential skills for working with data, statistics, and financial calculations.
To find a percent of a number, convert the percent to a decimal and multiply. For example, 30% of 80 is 0.30 × 80 = 24. You can also set up a proportion: 30/100 = x/80, so 100x = 2400 and x = 24.
This skill has many practical applications. Calculating a 15% tip on a $40 meal: 0.15 × 40 = $6. Finding a 20% discount on a $75 item: 0.20 × 75 = $15 off, so the sale price is $60. Computing the tithe: 10% of $500 income = $50. Being comfortable with percent calculations is essential for managing money wisely.
Percent change tells us how much a quantity has increased or decreased relative to its original value. The formula is: percent change = (amount of change / original amount) × 100. If a price goes from $80 to $100, the change is $20 and the percent increase is (20/80) × 100 = 25%.
Percent decrease works the same way. If enrollment drops from 200 to 170, the change is 30 and the percent decrease is (30/200) × 100 = 15%. Understanding percent change helps us interpret news reports, financial statements, and statistical data accurately — an important skill for being informed, discerning citizens.
Write thoughtful responses to the following questions. Use evidence from the lesson text, Scripture references, and primary sources to support your answers.
Solve this proportion: If 3 missionaries can reach 450 people in a week, how many people could 7 missionaries reach at the same rate? Set up and solve a proportion.
Guidance: Set up 3/450 = 7/x or equivalently 450/3 = x/7. Cross multiply and solve.
A family earns $4,200 per month. They give 10% to their church, save 15%, and spend 30% on housing. Calculate the dollar amount for each category. How much remains for other expenses?
Guidance: Find each percent of $4,200, then subtract all three from the total to find the remainder.
Jesus said that faithfulness with little leads to faithfulness with much (Luke 16:10). How is this like a mathematical proportion? What does this principle mean for how you handle responsibilities today?
Guidance: Think about proportional relationships — the ratio of faithfulness to responsibility stays constant regardless of scale. Apply this to schoolwork, chores, and relationships.