Math Notes

Basic Mathematics

Master foundational skills—arithmetic, fractions, ratios, percentages, and word problems. Ideal for Grades 1–12 and MTAP prep.

Solved Problems

Topics
MTAP - SOLUTIONS - Grade 10
A cylindrical bottle with radius 15cm is half-half filled with water. A spherical stone is dropped inside the bottle (completely submerging into the water) and the water level rises by 2.5 cm without spilling over. What is the radius of the stone?Chris drives from city A to City B on an expressway. After driving for 2.5 hrs, Chris saw a sign that reads City A-225 km: City B-198km. Continuing at the same constant speed, what is his total travel time from City A to City B?If 0.15a = 0.33b, what should be the value of x in the proportion a:bx = 11:10?Twice a certain Number is the average of \(\frac{7}{8}\) and \(1\frac{3}{4}\). What is the number? In a class, the ratio of the number of girls to boys is 5:4. If only 15 more girls had enrolled in the class, the ratio would have been 25:8. How many girls are in the class?What is the distance between \( -1\frac{1}{6}\) and \(-\frac{3}{4}\)?if \(f(x) = 2x^2 -1\) and \(g(x) = 8-2x\), what is the value of (f+g)(2)?If \(f(x) = x^2+2x-1\), what is f(2)?Find the values of x such that \(f(x) = 4\) in the equation \( f(x) = 2x^2 -3x +5\)
MTAP - SOLUTIONS - Grade 9
Solve for x in \( \frac{5}{x+2} - \frac{2}{x+1} = \frac{1}{2}\)Solve for x in \( \sqrt{2-x} -4 = 0\)Find the range of value of b if \(2x^2 +bx +8 = 0\) has no real roots. Express your answer in interval.List down all possible values of the constant k so that \( 3x^2 -2kx + 2 = 0\)Find the sum of the distinct roots of \( (x+\frac{2}{x})^2-6(x+\frac{2}{x})+9 = 0\)Solve the inequality \(2x^2 -7x -4 \leq 0 \). Express your answer in interval notation.Find the larger root of \(10x^2 -x = 3 \)What constant should be added to \( x^2 -10x\) to make it perfect square?Simplify \( \sqrt{72} +\sqrt{32}-\sqrt{2}\)Simplify \( (\frac{b^{4x-3y}}{b^{3x+2y}})^{x} (\frac{b^{3x+6y}}{b^{-2x+4y}})^y\)Rationalize the denominator of \(\frac{\sqrt{15} - 3}{\sqrt{5} -\sqrt{3}}\)Simplify \(\frac{3^0 - 3^{-2}}{3^{-1} + 3^{-3}}\)
MTAP - SOLUTIONS - Grade 8
Find the next two terms of the elements of the sequence 1, 8, 9, 32, 49, ...
Proving Identities
Show that: \(1 + {\left( {\cot \theta } \right)^2} = {\left( {\csc \theta } \right)^2}\)
Quadratic Equations
After plowing a border inside a rectangular field 60 rods wide and 80 rods long, a farmer finds that one half of the field remains to be plowed. How wide is the border?
Math Riddle
One rabbit saw six elephants while going to the river. <br> Every Elephant saw two monkeys going to the river. <br> Every monkey holds one parrot in their hands.<br> How many animals are going to the river?
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basic math elementary math MTAP reviewers grade 1 math grade 2 math grade 3 math grade 4 math grade 5 math grade 6 math junior high math senior high math word problems fractions ratios decimals percentages