1. Solving Problems Involving Formulas Many 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 Formulas We 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 = lw Triangle: P = a + b + c, A = bh Circle: C = πd, A = π r2
3. Problems using Distance formula How far can I go in 2 hours if I drive 75 miles per hour? FIND: distance FACTS: rate = 75, time = 2 FORMULA: D = rt (distance = rate * time) SUBSTITUTE: D = 75 • 2 SOLVE: D = 150 ANSWER: D = 150 miles
4. Solve the distance formula for other variables If 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 = r If Joe runs 4 miles in 20 minutes, what is his speed in mph? FIND: rate FACTS: Distance = 4, time = 20/60 or 1/3 of an hour FORMULA: r = D ÷ t SUBSTITUTE: r = 4 ÷ SOLVE: r = 4 • = 12 ANSWER: rate = 12 mph. Check. To find time, solve the formula for t: D = rt(divide both sides by r) D ÷ r = t How long does it take to drive 300 miles at 75 miles per hour? FIND: time FACTS: rate = 75, Distance = 300 FORMULA: t = D ÷ r SUBSTITUTE: t = 300 ÷ 75 SOLVE: t = 4 ANSWER: time = 4 hours, Check this answer in the original formula. Yes, it works.
5. Simple Interest Formula Find interest for $300 invested at 3% for 3 months FIND: amount of interest FACTS: Principle = 300, rate = .03, time = ¼ or .25 (Note: 3 months is 3/12 or ¼ of a year) FORMULA: I = prt SUBSTITUTE: I = 300 • .03 • .25 SOLVE: I = 2.25 ANSWER: amount of interest = $2.25.
6. Solve the Interest formula for p What if we need to find the principle rather than the amount of interest? Solve the formula for p: I = prt I ÷ (rt) = p Example: How much do I need to invest at 4% to earn $10 in 2 years? FIND: the principle FACTS: rate = .04, time = 2 FORMULA: p = I ÷ (rt) SUBSTITUTE: p = 10 ÷ (.04 • 2) SOLVE: p = 10 / .02 p = 500 ANSWER: I must invest $500. Substitute the values in the original formula to check.
7. Fahrenheit and Celsius Solve 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 w Let’s take the formula for perimeter of a rectangle and solve it for w P = 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)
