This document discusses using formulas to solve problems. It provides examples of common formulas for distance, rate, time, simple interest, Celsius to Fahrenheit conversions, and geometry. It demonstrates how to substitute values into formulas and how to solve formulas for different variables. For instance, it shows how to solve the distance formula for rate or time, the interest formula for principle, and the Fahrenheit formula for Celsius. Being able to manipulate formulas to solve for unknown values is an important problem solving skill.

1. Solving Problems Involving FormulasMany problems can be solved simply by substituting values into standard formulas. Others may require that we first solve for one of the variables in the formula.

2. Using FormulasWe have can use standard formulas for many types of problems, such as:Distance: D = rt (Distance = rate • time)Simple Interest: I = prt (Interest = principle • rate • time)Celsius to Fahrenheit: F = C + 32 (Fahrenheit = • Celsius + 32)Geometry: Rectangle: P = 2(l + w), A = lwTriangle: P = a + b + c, A = bhCircle: C = πd, A = π r2

3. Problems using Distance formulaHow far can I go in 2 hours if I drive 75 miles per hour?FIND: distance FACTS: rate = 75, time = 2FORMULA: D = rt (distance = rate * time)SUBSTITUTE: D = 75 • 2SOLVE: D = 150ANSWER: D = 150 miles

4. Solve the distance formula for other variablesIf the problem asks for rate or time we can solve our formula for that variable:To find a rate, solve the formula for r: D = rt(divide both sides by t) D ÷ t = rIf Joe runs 4 miles in 20 minutes, what is his speed in mph? FIND: rateFACTS: Distance = 4, time = 20/60 or 1/3 of an hourFORMULA: r = D ÷ tSUBSTITUTE: r = 4 ÷ SOLVE: r = 4 • = 12ANSWER: rate = 12 mph. Check. To find time, solve the formula for t: D = rt(divide both sides by r) D ÷ r = tHow long does it take to drive 300 miles at 75 miles per hour?FIND: timeFACTS: rate = 75, Distance = 300FORMULA: t = D ÷ rSUBSTITUTE: t = 300 ÷ 75SOLVE: t = 4ANSWER: time = 4 hours, Check this answer in the original formula. Yes, it works.

5. Simple Interest FormulaFind interest for $300 invested at 3% for 3 monthsFIND: amount of interestFACTS: Principle = 300, rate = .03, time = ¼ or .25 (Note: 3 months is 3/12 or ¼ of a year)FORMULA: I = prtSUBSTITUTE: I = 300 • .03 • .25SOLVE: I = 2.25ANSWER: amount of interest = $2.25.

6. Solve the Interest formula for pWhat if we need to find the principle rather than the amount of interest?Solve the formula for p: I = prtI ÷ (rt) = pExample: How much do I need to invest at 4% to earn $10 in 2 years?FIND: the principleFACTS: rate = .04, time = 2FORMULA: p = I ÷ (rt)SUBSTITUTE: p = 10 ÷ (.04 • 2)SOLVE: p = 10 / .02 p = 500ANSWER: I must invest $500. Substitute the values in the original formula to check.

7. Fahrenheit and CelsiusSolve the Fahrenheit formula for Celsius: We need to get C alone on one side of the equation. F = C + 32 F - 32 = C + 32 – 32 (subtract 32 from both sides) F - 32= C (F - 32) ÷ = C (divide both sides by ) (F - 32) • = C (change to multiplication by reciprocal) C = (F - 32) (commutative / symmetric properties)

8. Solve Perimeter Formula for wLet’s take the formula for perimeter of a rectangle and solve it for wP = 2(l + w) (divide both sides by 2) = l + w (cancel the factor of 2) - l = w (subtract l from both sides) w = - l (symmetry)