How to Solve Quadratic Word Problems Part 3

In the previous two posts (Part 1 and Part 2), we have discussed two problems involving quadratic equations. The first one is

A rectangular flower garden with dimensions 3m by 7m is surrounded by a walk of uniform width. If the area of the walk is 11 square meters, what is the width of the walk in meters?

Solution

If we let x be the width of the walk, then the length of the walk becomes x + 7 + x = 7 + 2x and the width becomes x + 3 + x = 3 + 2x as shown in the figure below.

cs2

We know that the area of the garden (the inner rectangle) is equal 7 \times 3 = 21 square meters. The area of the larger rectangle is equal to

(3 + 2x)(7 + 2x) = 21 + 14x + 6x + 4x^2 = 21 + 20x + 4x^2.

Now, to get the area of the walk, we have to subtract the area of the garden from the area of the large rectangle. That is,

area of the larger rectangle – area of the smaller rectangle = 11

4x^2 + 20x + 21 - 21 = 11

4x^2 + 20x = 11

Subtracting 11 from both sides, we have

4x^2 + 20x -11 = 0.

Now, the quadratic equation is not factorable, so we use the quadratic formula in order to find the value of x. In using the quadratic formula, we want to identify the values of a, b, and c in the equation ax^2 + bx + c = 0 and substitute in the quadratic formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

From the equation above, a = 4, b = 20 and c = -11. Substituting these values in the quadratic formula, we have

x = \dfrac{-20 \pm \sqrt{(-20)^2 - 4(4)(-11)}}{2(4)}

= \dfrac{-20 \pm \sqrt{400 + 176}}{8}

= \dfrac{20 \pm 24}{8}

That means that we have two roots

x_1 = \dfrac{-20 + 24}{8} = \dfrac{1}{2}
and

x_2 = \dfrac{-20 - 24}{8} = \dfrac{-44}{8} = \dfrac{-11}{2}.

As we can see, we x_2 is not a possible root because it is negative. Therefore, the acceptable answer to the problem is

x_1 = \dfrac{1}{2}.

This means that the width of the walk is ½ meters.

Check: If the width of the walk is ½ meters, then the length of the larger rectangle is 1/2 + 7 + 1/2 = 8 and the width is 1/2 + 3 + 1/2 = 4. The area is (8)(4) = 32 sq. m.

Now the area of the walk is 21 meters and 32 - 21 = 11 sq. m. which is the area of the walk indicated in the problem. Therefore, we are correct.

You can also watch the Youtube video below regarding the discussion above.

Enjoy!

Downloadable Math Word Problem Tutorials

Many of you had been asking for downloadable PDF of tutorials. Well, this is what you’ve been waiting for! Admin Jess Christian Ramirez has prepared some tutorials on math word problems with examples which you can download in PDF form. You may download them, print them, and share them with your friends!

Also, you might want to join our Facebook Group. We have more than 10,000 members reviewing through question and answer about Civil Service Exams.

0.) Basic Mathematics

1.) Number Relation Problems

2.) Geometry Word Problems 

3.) Money Problems

4.) Age Problems

5.) Work Problems

6.) Investment Problems

7.) Mixture Problems

8.) Motion Problems

Enjoy studying and good luck every one.

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Introduction to Venn Diagrams

According to some examinees, there are several items in the October 2014 Civil Service Exam that requires the use of Venn diagrams, so I will discuss in details how to use Venn diagrams in a series of posts.  In this post, I am going to introduce its concept and its use as layman as possible. I will limit the discussion to the preparation of solving word problems involving Venn diagrams.

Venn diagrams are used to represent logical relations between sets. In word problems involving diagrams, elements of sets are usually people who choose or prefer a particular thing (e.g. color, food, hobbies). For instance, we have two available desserts, biscuits and cookies, and then Abby ate a biscuit and Bella ate a cake. Chubby, being extremely hungry, ate a biscuit and a cake. So, the situation can be represented as follows using a Venn diagram.  » Read more

How to Solve Number Series Tutorials

The “number series” tutorials of PH Civil Service review contains tutorials on methods and strategies used in solving number series problems. As I have mentioned in the teaser of this tutorials, the term “series” is technically wrong: what you are solving are really sequences. Although they maybe similar to many of you, to mathematics majors, they are very different.  In mathematics, series means sequence of sums. I will not go into details about this since this is irrelevant for the review. In this series of tutorials, mathematically, we will use the term “series” and “sequence” interchangeably.

1.) A Teaser on Answering Number Series Questions discusses a brief introduction to number sequence.

2.) How to Solve Number Series Problems Part 1 discusses simple integer, letter, and fraction sequence.  » Read more

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