Worked Problems (Modeling With Equations)

September 4th, 2009

1. Find four consecutive odd integers whose sum is 272.

Solution

Let be the smallest numbers among these four numbers. Then the four numbers are . We have

So the smallest among the four is 65. Therefore the four numbers are .

2. What annual rate of interest would you have to earn on an investment of $3500 to ensure receiving $262.50 interest after one year?

Solution

Let the interest rate be . The annual interest earned at this rate is and we want to make this equals . So we have an equation that

Multiply both sides by 100 we have . Therefore . Thus the rate is .

3. A woman earns 15% more than her husband. Together they make $69,875 per year. What is the husband’s salary?

Solution

Let be the husband’s salary. Then the salary of his wife is . Together their salary is

Therefore . Hence the salary of the husband is.

4. A father is four times as old as his daughter. In 6 years, he will be three times as old as she is. How old is the daughter now?

Spoiler

Let be the age of the daughter now. Then his father’s age is . In 6 years the age of the daughter and the father are and respectively. We also know that in 6 years the father is 3 times old as the daughter is. Hence

So the daughter is 12 years old now.

5. A pasture is twice as long as it is wide. Its area is 115,200 . How wide is the pasture?

Solution

Let be the width of the pasture. Since the length is twice of it is width. So the length is . It follows that the area is equal to
and at the same time equals 115,200. So then we have . Divide both sides by 2 we have . Therefore .

6. Two cyclists, 120 mi apart, start riding toward each other at the same time. One cycles twice as fast as the other. If they meet 2 hour later, at what average speed is each cyclist traveling?

Solution

Let the speed of the first and the second cyclist be and respectively.
In two hours the first cyclist traveled for miles and the second cyclist traveled for miles.
Then we have . Therefore and hence the average speed of the first and the second cyclist are 20 mph and 40 mph respectively.

7. A rectangular parcel of land is 50 ft wide. The length of a diagonal between opposite corners is 10 ft more than the length of the parcel. What is the length of the parcel?

Solution

Let be the length of the parcel. Then the diagonal is . The length, the width and the diagonal forms a right triangle. So by Pythagorean’s Theorem we have . It follows that . Subtracting from both sides, we have . Hence and therefore ft.

8. Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

Solution

Suppose Stan can mow the lawns unit per minute. Then Hilda can lawn for per minute. Then in 40 minutes they can mow
unit. This means that if Stan work by himself, to get unit he needs to work for 120 minutes.

9. What quantity of 60% acid solution must be mixed with a 30% solution to produce 300mL of a 50% solution?

Solution

Let mL be the quantity of the 60% acid solution and mL be the quantity of the 30% acid solution. If we mix the solution we get mL solution. It follows that mL.
The amount of pure acid in the 60% solution is , in the 30% solution is and in the 50% solution is mL. So now we have

Therefore we need 200mL of 60% acid solution and mL of 30% solution to produce 300mL of a 50% solution.

10. A merchant blends tea that sells for $3.00 a pound with tea that sells for $2.75 a pound to produce 80 lb of a mixture that sells for $2.90 a pound. How many pounds of each type of tea does the merchant use in the blend?

Solution

Let the weight of the $3.00 a pound tea and $2.75 a pound be lb and lb respectively. Since the mixture of the two type produce 80lb of tea, then , hence . The first type of tea sells for dollars, the second type of tea sells for and the mixture sells for Therefore we have

It follows that .

Categories: Algebra, MA109

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