Word Problem - Number Problems, 6

Category: Algebra

"Published in Newark, California, USA"

The product of two consecutive integers is 106 less than twice the square of the next larger integer. Find the numbers.

Solution:

The first thing that we need to do is to analyze the given problem as follows:

Let x = be the first consecutive integer
      x + 1 = be the second consecutive integer
      x + 2 = be the third consecutive integer

If the word problem says "The product of two consecutive integers is 106 less than twice the square of the next larger integer.", then the working equation is
 

From the working equation, the value of x which is the first consecutive integer is
 

 

 

 

 

If you equate each factor to zero, then the values of x are -14 and 7.

Therefore, the three consecutive integers are -14, -13, and -12 or 7, 8, and 9.

Word Problem - Area Problems

Category: Algebra

"Published in Newark, California, USA"

In 1964, Thelma bought two lots at ₱ 9,600 each. One lot contains 40 square meters more than the other but costs ₱ 12 less per square meter. Find the area of each lot.

Solution:

The given problem is about finding the area of two lots in which Thelma bought.  The other lot has a bigger area but cheaper price. The prices of both lots are the same. Let's analyze the given word problem as follows:

Let x = be the area of Lot #1
      y = be the price of Lot #1 per square meter
      x + 40 = be the area of Lot #2
      y - 12 = be the price of Lot #2 per square meter

The working equation for Lot #1 is

The working equation for Lot #2 is

But  

Hence, the above equation becomes

By using the completing the square method, the value of x is

Since area is always a positive value, then we need to choose the positive sign. Therefore, the area of Lot #1 is

and the area of Lot #2 is 160 m² + 40 m² = 200 m². 

Square, Rectangle, and Parallelogram Problems, 11

Category: Algebra, Plane Geometry

"Published in Newark, California, USA"

Prospero bought a rectangular lot in Quezon City. If he needed 78 meters of chicken wire to fence the four sides inclosing 368 square meters of land, what are the dimensions of the lot?

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Let x = be the length of the rectangular lot 
      y = be the width of the rectangular lot
     
If the word statement says "If he needed 78 meters of chicken wire to fence the four side...", then the working equation is

If the word statement says "...to fence the four sides inclosing 368 square meters of land...", then the working equation is

But  

Hence, the above equation becomes

By using quadratic formula, the value of x is

If you consider the positive sign, the value of x is

and the value of y is

If you consider the negative sign, the value of x is 

and the value of y is

Therefore, the dimensions of the rectangular lot are 23 m by 16 m.