Fact: Math teachers use the the word "applications" as a euphemism for "word problems."
Rough Guide for Solving Word Problems :
- Step 1. Read the problem carefully to determine what you are being asked to find.
- Step 2. Select a variable to represent each unknown quantity. Specify precisely what each variable represents and note any restrictions on each variable.
- Step 3. If necessary, make a sketch and translate the problem into a word equation or a system of word equations. Then translate each word equation into an algebraic equation.
- Step 4. Solve the equation or system of equations, and answer the question completely in the form of a sentence.
- Step 5. Check the reasonableness of your answer.
A Unifying Principle: The Mixture Principle
Mixture Principle for Two Ingredients:
Amount Stuff in First + Amount Stuff in Second=Amount Stuff in Mixture
A Classic Mixture Example: Billy Bob wants to fortify his beer with some bourbon which is 100 proof (50 percent alchohol by volume). How many ounces of bourbon does Billy Bob need to add to his 22 oz beer which is 4 percent alcohol in order to get a fortified mixture which is 8 percent alcohol? Round your answer to the nearest tenth of an ounce.
Mixture Principle Says: The amount of alcohol before mixing is the same amount after mixing.
DISTANCE$=$RATE$\times$TIME
Example:Two boats leave a dock at the same time. One travels downstream at a rate of 16 mi/h and the other travels upstream at a speed which is 5 mi/h slower than the first. Determine the number of hours it will take the boats to be 60 mi apart (along the river). If necessary, round your answer to the nearest tenth.
Mixture Principle Says: The distance both boats travel individually is the same as the total distance.
Simple Interest: $I=Prt$
Example: A retiree receives a yearly income of $\$$10100 from her IRA to help fund her retirement. She has placed $\$$70000 of this account in a secure Treasury bond earning 4 percent yearly interest. If she earns a 8 percent rate of return on the rest of this IRA through an insurance annuity, what is total amount invested in the IRA? Round your answer to the nearest dollar.
Mixture Principle Says: The amount of interest each investment earns individually is the same as the total interest.
Bonus: In the above problem, what was the principal that is invested in the insurance annuity?