 # Maths Problem-solving Examples With Solutions

What are some examples of problem-solving strategies used in mathematics?

Math can be studied independently if you can comprehend simple English and have access to the internet. You’ll discover that you are the only person who can teach yourself faster and better after putting everything in this manual into practice. After you implement everything in this guide, you’ll discover that there’s no one who can understand maths faster and better than you.

Just a small word of caution: even though I said anyone can do this, I’m absolutely certain not everyone will. Actually, it’s a little unsettling, particularly if you’re doing this for the first time.

As you work through a four-step problem-solving process, elementary math students using the “Four-Step Problem Solving” plan are encouraged to use sound reasoning and develop their mathematical language. The details, main idea, strategy, and how steps make up this problem-solving plan.

Students may use “graphic representations” as they progress through each stage to arrange their thoughts, show how they think mathematically, and demonstrate how they plan to solve a problem.

## Main Idea

The student performs the roles of reader, thinker, and analyst in this step. The student begins by reading the article and scanning for any proper words (words with insight). If unusual names of people or places cause confusion, the student may substitute a familiar name and see if that is the problem. Now the puzzle is clear. Reread the issue, summarizing it, or putting the issue into a visual form may be helpful to the student.

## Details

The student should read the issue once more, word by word, slowly and attentively. By utilizing words, numbers, and phrases, you would recognize and document any details. The student searches for additional information or facts from the reading that is related to the answer. The student should continue to search for any hidden numbers that may be implied but not explicitly stated in this step.

For instance: The issue might concern “Frank and his three friends.” (In order to solve the problem, the student needs to understand that there are actually four people, even though “four” or “4” is not mentioned in the reading.)

## Strategy

The student picks a math strategy (or several strategies) to use in order to solve the problem and finds the solution using that strategy. The following are some potential tactics, according to the Texas Essential Knowledge and Skills (TEKS) curriculum.

• Use or create a visual
• Observe a pattern
• Construct a number of sentences.
• Use procedures (operations) like addition, subtraction, multiplication, and division
• Create or utilize a table
• Use or create a list.
• Tto solve a simpler issue
• How

Students use words or phrases to explain how they came up with their solution in order to make sure that it is reasonable and that they fully comprehend the process. The following are some queries that students should ask themselves.

• What was the solution to the issue?
• How did I choose my approach?
• “What steps did I take?”

A student explanation of the chosen solution approach is required in this step. They need to justify their course of action and offer evidence that it is sound. From this step, students have the chance to show how they understood the mathematical ideas and vocabulary used in the problem they solved.

1. In comparison to the morning, a salesman sold twice as many pears in the afternoon. If he sold 360 kilograms of pears that day, how many kilograms did he sell in the morning, and how many in the afternoon?

Solution:

Let x

x is the number of kilograms he sold in the morning.

Then in the afternoon, he sold

2x

2x kilograms. So, the total is

x + 2x = 3x

x+2x=3x. This must be equal to 360.

3x = 360

3x=360

x=3/360

x = 120

x=120

Therefore, the salesman sold 120 kg in the morning and

In the afternoon, he sold 2 X 120 = 240kg.

2. Mary, Peter, and Lucy were out picking chestnuts. Compared to Peter, Mary picked twice as many chestnuts. Peter picked up 2 kg less than Lucy did. The three of them collectively collected 26 kg of chestnuts. How many kilograms did each of them pick?

Solution:

Let x be the amount Peter picked.

Then Mary picked 2x

Lucy picked x+2 respectively.

So x+2x+x+2=26.

4x=24

x=24/4

x=6

Therefore, Peter, Mary, and Lucy picked 6, 12, and 8 kg, respectively.

3. An airplane that flies against the wind from A to B in 8 hours. The same aircraft takes seven hours to fly back from B to A in the same direction as the wind. Calculate the difference between the airplane’s speed in still air and the wind’s speed.

Let x = the speed of the airplane in still air, y = the speed of the wind, and D = the distance between A and B. Find the ratio x / y

Against the wind: D = 8(x – y), with the wind: D = 7(x + y).

8x – 8y = 7x + 7y, hence x / y = 15.

4. Find the area between two concentric circles defined by
x2 + y2 -2x + 4y + 1 = 0
x2 + y2 -2x + 4y – 11 = 0

Solution:

Rewrite equations of circles in standard form. Hence equation

x 2 + y2 -2x + 4y + 1 = 0 may be written as

(x – 1)2 + (y + 2) 2 = 4 = 22

and equation

x2 + y2 -2x + 4y – 11 = 0 as

(x – 1)2 + (y + 2) 2 = 16 = 42

Knowing the radii, the area of the ring is π (4)2 – π (2)2 = 12

5. Find all values of parameter m (a real number) so that the equation 2x2 – m x + m = 0 has no real solutions.

The given equation is a quadratic equation and has no solutions if its discriminant D is less than zero.

D = (-m)2 – 4(2)(m) = m2 – 8 m

We nos solve the inequality m2 – 8 m < 0

The solution set of the above inequality is: (0 , 8)

Any value of m in the interval (0, 8) makes the discriminant D negative and, therefore, the equation has no real solutions.

6. The sum of an integer N and its reciprocal is equal to 78/15. What is the value of N?

This equation should be written in N.

N + 1/N = 78/15

Multiply all terms by N, obtain a quadratic equation, and solve to obtain N = 5.

7. M and N are integers, so that 4m / 125 = 5n / 64. Find values for m and n.

4m / 125 = 5n / 64

Cross multiply: 64 (4)m = 125 (5n)

Note that 64 = 4(3) and 125 = 5 (3)

The above equation may be written as: 4m + 3 = 5n + 3.

The only values of the exponents that make the two exponential expressions equal are: m + 3 = 0 and n + 3 = 0, which gives m = – 3 and n = – 3

The summary

If you believe that you are not a “math person,” you would require a different individual to teach you math in a classroom setting. Let me tell you this. With all the free resources available online, including lectures, syllabi, ebooks, and MOOCS, it is possible to self-study math with relative ease, just like you would in a college setting.

What’s great is that you can go at your own pace. No strict schedules, just self-commitment If you want to benefit, you must reconsider your perspective on this, though. You must bear in mind that the mental effort you spend mastering a math concept is the cost you must bear in exchange for future math skills becoming simpler or, more accurately, it’s the cost you incur to ensure that you don’t make learning challenging for yourself in the future.