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. 

Rate, Distance, and Time Problems, 9

Category: Mechanics, Physics

"Published in Newark, California, USA"

Felix rode 18 miles downstream and back to the same place where he started, in 3 hours and 12 minutes. Find the rate of Felix in still water if the velocity of the stream is 3 miles per hour.

Solution:

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

Photo by Math Principles in Everyday Life

Let s = be the distance traveled by Felix in upstream / downstream
      v = be the velocity or speed of Felix in still water
      v₁ = be the velocity in downstream
      v₂ = be the velocity in upstream 
       t₁ = time traveled in downstream
       t₂ = time traveled in upstream 

If the word statement says "Felix rode 18 miles downstream and back to the same place where he started, in 3 hours and 12 minutes.", then the working equation is

Substitute the values of t₁ and t₂ so that we can solve for the value of v which is Felix' speed or velocity in still water. 

By using quadratic formula, the value of v is 

We need to choose the positive sign not the negative sign because the value of velocity will be negative which is not accepted as a final answer.  Therefore, the velocity or speed of Felix in still water is