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# Problems Involving Formulas

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Solve Word Problems using standard formulas

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### Problems Involving Formulas

1. 1. Solving Problems Involving Formulas<br />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.<br />
2. 2. Using Formulas<br />We have can use standard formulas for many types of problems, such as:<br />Distance: D = rt (Distance = rate • time)<br />Simple Interest: I = prt (Interest = principle • rate • time)<br />Celsius to Fahrenheit: F = C + 32 <br /> (Fahrenheit = • Celsius + 32)<br />Geometry: <br />Rectangle: P = 2(l + w), A = lw<br />Triangle: P = a + b + c, A = bh<br />Circle: C = πd, A = π r2<br />
3. 3. Problems using Distance formula<br />How far can I go in 2 hours if I drive 75 miles per hour?<br />FIND: distance <br />FACTS: rate = 75, time = 2<br />FORMULA: D = rt (distance = rate * time)<br />SUBSTITUTE: D = 75 • 2<br />SOLVE: D = 150<br />ANSWER: D = 150 miles<br />
4. 4. Solve the distance formula for other variables<br />If the problem asks for rate or time we can solve our formula for that variable:<br />To find a rate, solve the formula for r: <br /> D = rt(divide both sides by t)<br /> D ÷ t = r<br />If Joe runs 4 miles in 20 minutes, what is his speed in mph? <br />FIND: rate<br />FACTS: Distance = 4, time = 20/60 or 1/3 of an hour<br />FORMULA: r = D ÷ t<br />SUBSTITUTE: r = 4 ÷ <br />SOLVE: r = 4 • = 12<br />ANSWER: rate = 12 mph. Check.<br /> To find time, solve the formula for t:<br /> D = rt(divide both sides by r)<br /> D ÷ r = t<br />How long does it take to drive 300 miles at 75 miles per hour?<br />FIND: time<br />FACTS: rate = 75, Distance = 300<br />FORMULA: t = D ÷ r<br />SUBSTITUTE: t = 300 ÷ 75<br />SOLVE: t = 4<br />ANSWER: time = 4 hours, Check this answer in the original formula. Yes, it works.<br />
5. 5. Simple Interest Formula<br />Find interest for \$300 invested at 3% for 3 months<br />FIND: amount of interest<br />FACTS: Principle = 300, rate = .03, time = ¼ or .25 (Note: 3 months is 3/12 or ¼ of a year)<br />FORMULA: I = prt<br />SUBSTITUTE: I = 300 • .03 • .25<br />SOLVE: I = 2.25<br />ANSWER: amount of interest = \$2.25. <br />
6. 6. Solve the Interest formula for p<br />What if we need to find the principle rather than the amount of interest?<br />Solve the formula for p: I = prt<br />I ÷ (rt) = p<br />Example: How much do I need to invest at 4% to earn \$10 in 2 years?<br />FIND: the principle<br />FACTS: rate = .04, time = 2<br />FORMULA: p = I ÷ (rt)<br />SUBSTITUTE: p = 10 ÷ (.04 • 2)<br />SOLVE: p = 10 / .02<br /> p = 500<br />ANSWER: I must invest \$500. Substitute the values in the original formula to check.<br />
7. 7. Fahrenheit and Celsius<br />Solve the Fahrenheit formula for Celsius: We need to get C alone on one side of the equation.<br /> F = C + 32 <br /> F - 32 = C + 32 – 32 (subtract 32 from both sides)<br /> F - 32= C <br /> (F - 32) ÷ = C (divide both sides by ) <br /> (F - 32) • = C (change to multiplication by reciprocal)<br /> C = (F - 32) (commutative / symmetric properties)<br />
8. 8. Solve Perimeter Formula for w<br />Let’s take the formula for perimeter of a rectangle and solve it for w<br />P = 2(l + w)<br /> (divide both sides by 2)<br /> = l + w (cancel the factor of 2) <br /> - l = w (subtract l from both sides)<br /> w = - l (symmetry)<br />