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.

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

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

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 = 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.

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.

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 = 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.

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)

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)