The Mathematics Teachers Association of the Philippines, Inc. (MTAP) is one of my most awaited events before. Personally, I have loved joining math contests since I was young. A part of joining those events gave me valuable experiences that I consider one of the foundations that let me develop a great passion for numbers and math, itself.
In this article, I have come up with some questions with detailed solutions where you can learn some techniques on what approach you can apply to solve such problems. The first set of questions are made for grade 7 and 8 students. The latter part is for grade 9 and 10 students.
So if you are in middle school and having a little struggle understanding some math questions, a few selected problems pertaining to some common math problems are presented.
Problems involving simplifying of terms and fractions
Problems involving simplifying of terms and fractions
Simplify: 5(2)^3 − (6 − 8)^2.
Solution:
5(2)^3 − (6 − 8)^3
5(8) - (-2)^3
40 - (-8)
40 + 8
= 48
Hence, the answer is 48.
By how much is 2 − 1/2 greater than 1/2 − 4?
Solution:
(2 - 1/2) - (1/2 - 4)
(4/2 - 1/2) - (1/2 - 8/2)
(3/2) - (-7/2)
3/2 + 7/2
=9/2
The answer is 9/2.
The average of four numbers is 36. Two numbers are added to the set, and the average of the six numbers becomes 57. If one of the added numbers is twice the other, what are the two numbers added to the data set?
Solution:
n = 4(36) = 144.
Let x and y be the two numbers added to the set.
n + x + y = 6(57) = 342
342 - 140 = x + y
198 = x + y
x = 2y
2y + y = 198
3y = 198
y = 66
x = 198 - y
x = 198 - 66
x = 132
The two numbers are 66 and 132.
Two prime numbers have a product of 302. Find the sum of the two numbers.
Solution:
Since 302 is an even number, one of the factors is 2.
(302 ÷ 2) + 2
=153
You will get 153.
Problems involving more complex computations
Subtract 4x − 3y + 2z from the sum of 5x + 2y − z and 2x − 2y + 5z.
Solution:
[(5x+2y-z)+(2x-2y+5z)]−(4x−3y+2z)=(7x+4z)−(4x-3y+2z)
=7x+4z-4x+3y-2z
=3x+3y-6z
The final answer is 3x+3y-6z.
The sum of the measures of the interior angles of a polygon is 1800◦. How many sides has the polygon?
Solution:
180(n − 2) = 1800
180n - 360= 1800
180n = 1800 + 360
180n = 2160
n=12
The polygon has 12 sides.
Problems involving True or False Questions
If n is a real number, then n^2 is positive.
Explanation:
If n = 0, n^2 = 0 which is neither positive nor negative.
Hence, the statement is false.
If three children eat 4 kilos of rice in 5 days, how long will 12 children eat 48 kilos of rice? Is it 15 days?
Solution:
Divide 4 kilos of rice to the number of children to determine how much each of these children ate in 5 days. After that, divide it again by 5 to get the daily intake of rice by each of these children.
= (4/3)/5
= 0.2667
Now, multiply the result by the number of new children. The answers will then be divided to the number of rice in kilograms.
= 0.2667
Now, multiply the result by the number of new children. The answers will then be divided to the number of rice in kilograms.
= (12x0.2667)
= 3.2
= 3.2
= 48/3.2
= 15 days
The answer for this statement is true.
Logic and Age Problems
The number of boys in a class is equal to the number of girls. Ten boys are absent today, and this leaves thrice as many girls as boys in the classroom. How many students belong to the class?
x = 3(x − 10)
x = 3x - 30
30 = 3x - x
30 = 2x
15 = x
Total students = 2x
= 2(15)
= 30
The number of students that belong in the class is 30.
Hanz is 8 years old and Carl is 36. In how many years will Carl be 3 times as old as Hanz?
Solution:
36 + x = 3(8 + x)
36 + x = 24 + 3x
36 - 24 = 3x - x
12 = 2x
6 = x
Hence, it is 6 years.
Now, let us move on to Grade 9 and 10 Mathematics. These Questions and Solutions are specially made to have a clearer understanding to these topics.
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Solution:
Square both sides of the equation:
c^2 = a^2 + b^2
2^2 = 1^2 + b^2
4 = 1 + b^2
4-1 = b^2
3 = b^2
b= square root of 3
Answer: √3 m (square root of 3 m)
Arithmetic
A sequence is defined by an = 3(an−1 + 2) for n ≥ 2, where a1 = 1. What is a4?
Solution:
a2 = 3(a1 + 2) = 9
a3 = 3(a2 + 2) = 33
a4 = 3(a3 + 2) = 105
The fourth term is 105.
A worm crawls 7.5 inches in 80 seconds. What is its speed in feet per hour?
Solution:
speed = 7.5 in/80s
s = 7.5 in/80s
s = 0.09375 in/s
s = 0.09375 x 3,600 s
s= 337.5 in/hr
s = 337.5 in/12 n
s= 28.125 ft/hr
The speed is 28.125 ft/hr.
Logic and Age Problem
Solution:
36 + x = 3(8 + x)
36 + x = 24 + 3x
36 - 24 = 3x - x
12 = 2x
6 = x
Hence, it is 6 years.
Now, let us move on to Grade 9 and 10 Mathematics. These Questions and Solutions are specially made to have a clearer understanding to these topics.
Simplifying of terms
Simplify 
Solution:
Rationalize the denominator:
Therefore, the answer is 
Find two positive consecutive integers whose product is 506.
Solution:
Let x is equal to the smaller number and x + 1 be the larger number.
x(x+1)=506
x^2+x=506
x^2+x-506=0
(x-22)(x-23)
x=22, 23
Solution:
Let x is equal to the smaller number and x + 1 be the larger number.
x(x+1)=506
x^2+x=506
x^2+x-506=0
(x-22)(x-23)
x=22, 23
The two positive consecutive integers are 22 and 23.
Solve for 
in 
.
in .Solution:
Square both sides of the equation:
Square both sides:
Hence, the answer is 36.
Evaluate 
.
.Solution:
The answer is 4.
Finding the value of x
Solve for x in x^2 + 12 < 7x.
Solution:
x^2 + 12 < 7x
x^2 − 7x + 12 < 0
(x − 4)(x − 3) < 0
x ∈ (3, 4)
Therefore, x ∈ (3, 4)
Find all possible values of x if one of the interior angles of a square is (2x 2 − 8x)◦.
The possible values of x are −5 and 9.
Solution:
2x^2 − 8x = 90
Divide both sides by 2
x^2 − 4x = 45
x 2 − 4x − 45 = 0
(x − 9)(x + 5) = 0
x = 9, -5
The possible values of x are −5 and 9.
Geometry-related questions
What is the height of an equilateral triangle whose perimeter is 6 meters?
Solution:
The side measures 2 m.c^2 = a^2 + b^2
2^2 = 1^2 + b^2
4 = 1 + b^2
4-1 = b^2
3 = b^2
b= square root of 3
Answer: √3 m (square root of 3 m)
A sequence is defined by an = 3(an−1 + 2) for n ≥ 2, where a1 = 1. What is a4?
Solution:
a2 = 3(a1 + 2) = 9
a3 = 3(a2 + 2) = 33
a4 = 3(a3 + 2) = 105
The fourth term is 105.
A worm crawls 7.5 inches in 80 seconds. What is its speed in feet per hour?
Solution:
speed = 7.5 in/80s
s = 7.5 in/80s
s = 0.09375 in/s
s = 0.09375 x 3,600 s
s= 337.5 in/hr
s = 337.5 in/12 n
s= 28.125 ft/hr
The speed is 28.125 ft/hr.
Logic and Age Problem
Andy is 4 years old and Richard is 32. In how many years will Charlie be 3 times as old as Andy?
Solution:
32 + n = 3(4 + n)
32 + n = 12 + 3n
32 - 12 = 3n - n
20 = 2n
10 = n
The answer is 10 years.
Solution:
32 + n = 3(4 + n)
32 + n = 12 + 3n
32 - 12 = 3n - n
20 = 2n
10 = n
The answer is 10 years.
These are just a few of the commonly asked math problems. They may look chanllenging but once you know the process on how you can come up with their correct answers, you will for sure enjoy doing it.
Should you have encountered errors or mistakes, feel free to write them in the comment box, or you may simply use the contact page to address those errors with us.
Do not fear math. Instead love it by practicing more often on solving math-related questions just like what you just have read. While on the process, do not forget to celebrate small wins because every achievement, no matter how small they are, are worth celebrating.
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