Lesson 2.2: Use a Problem Solving Strategy

New Concept

Learn a powerful, universal strategy to conquer any word problem. This method breaks down challenges into simple steps, helping you translate real-world scenarios involving numbers, percents, and simple interest into easily solvable equations.

What’s next

This strategy is your new toolkit. Next, you’ll master it step-by-step through worked examples, interactive practice cards, and challenge problems.

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Use a Problem Solving Strategy for word problems.

• Step 1. Read the problem. Make sure all the words and ideas are understood.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
• Step 5. Solve the equation using proper algebra techniques.
• Step 6. Check the answer in the problem to make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

Examples

• The number of apples in a basket is 5 more than twice the number of oranges. If there are 31 apples, how many oranges are there? Let $o$ be the number of oranges. The equation is $2o + 5 = 31$. Solving gives $2o = 26$, so $o = 13$. There are 13 oranges.

• The sum of five times a number and ten is 55. Find the number. Let $n$ be the number. The equation is $5n + 10 = 55$. Solving gives $5n = 45$, so $n = 9$. The number is 9.

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Consecutive integers are integers that immediately follow each other. If the first integer is $n$, the next are $n+1, n+2, …$

Consecutive even integers are even integers that immediately follow one another. If the first even integer is $n$, the next are $n+2, n+4, …$

Consecutive odd integers are odd integers that immediately follow one another. If the first odd integer is $n$, the next are $n+2, n+4, …$

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To solve percent equations, translate English sentences into algebraic equations. Be sure to change the given percent to a decimal before you use it in the equation.

(a) What number is 45% of 84? translates to $n = 0.45 ⋅ 84$.

(b) 8.5% of what amount is 4.76 dollars? translates to $0.085 ⋅ n = 4.76$.

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To find the percent change:

• Step 1. Find the amount of change.

• Step 2. Find what percent the amount of change is of the original amount. The amount of change is what percent of the original amount?

Examples

• A shirt's price increased from 20 dollars to 25 dollars. Find the percent change. The change is $25 - 20 = 5$ dollars. The percent change is $⅔{5}{20} = 0.25$, which is a 25% increase.

• A car's value decreased from 15,000 dollars to 12,000 dollars. Find the percent change. The change is $12,000 - 15,000 = -3,000$ dollars. The percent change is $⅔{-3000}{15000} = -0.20$, a 20% decrease.

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Discount

Mark-up

Examples

• A coat with an original price of 200 dollars is on sale for 30% off. Find the sale price. The discount is $0.30 ⋅ 200 = 60$ dollars. The sale price is $200 - 60 = 140$ dollars.

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If an amount of money, $P$, called the principal, is invested or borrowed for a period of $t$ years at an annual interest rate $r$, the amount of interest, $I$, earned or paid is:

where $I$ = interest, $P$ = principal, $r$ = rate, and $t$ = time.

Examples

• Calculate the simple interest earned on 1,500 dollars at a 5% rate for 4 years. Using $I = Prt$, we get $I = (1500)(0.05)(4) = 300$. The interest earned is 300 dollars.

• Sam borrowed 4,000 dollars and paid 800 dollars in simple interest over 5 years. What was the interest rate? Using $I = Prt$, we have $800 = (4000)r(5) ➔ 800 = 20000r ➔ r = 0.04$. The rate was 4%.