Mathematics Concepts and Problems

Inverse Proportionality

• X and Y are inversely proportional if their product is constant.
• If X = 15,000 when Y = 162,500, find X when Y = 328,400.
• (X1)(Y1) = (X2)(Y2)
• (15000)(162500) = (X2)(328400)
• X2 = 7,422.35

Consecutive Integers

• The sum of seven consecutive integers is zero.
• Find the smallest integer.
• The smallest integer is -4.

Three Distinct Positive Integers

• The sum and product of three distinct positive integers are 15 and 45 respectively.
• Find the largest integer.
• The largest integer is 9.

Spherical Segment Curved Surface Area

• Find the curved surface area of a spherical segment.
• The diameters of the bases are 100 cm and 140 cm, and they are 25 cm apart.
• The curved surface area is 11,673.43 cm².

Map Scale

• The area of a park on a map is 500 mm²; the scale is 1:40,000.
• Determine the true area in hectares.
• The true area is 80 hectares.

Integral Evaluation

• Evaluate the integral ∫cscx cotx dx.
• The result is -cscx + C.

Differential Equation

• Find the general solution for y + 3y − 4y = 0.
• The solution is y = C₁e⁴ˣ + C₂e⁻ˣ

Ohm's Law

• Ohm's law states that the current in a resistor is proportional to the voltage.
• If 30 volts yields 2.4 x 10⁻³ amperes, find the constant of proportionality
• The constant of proportionality is 8 x 10⁻⁵.
• If voltage is 52 volts, the current is 4.16x10⁻³ amperes.

Linear Equation

• Find the value of x-y from 3x - 5y = 14; 6x - 3y = 21
• The value of x - y is 4.

Solving for x

• Solve for x if 48¹/₄ = 4 x ³/₂
• x = 2

Solving for x & y

• The equations are axb = 8, axc=3, bxc =6
• The value of x/y is 3/2

Non-zero solution

• Find the non-zero solution to 3x² - 30x = 0
• The non-zero solution is 10.

Possible Line Length

• A line is divided into 10 equal parts.
• The measure of each part is a prime integer.
• Find the possible length of the line.
• The possible length is 150.

Hat Problem

• Mano's hat count is four more than Alex's hat and one half that amount of Miguel's.
• The total number of hats is 24.
• Find Miguel's hats.
• Miguel has 16 hats.

Ratio of Portions

• The shipment of items is divided into two portions.
• The difference is half the average of the two portions.
• The ratio of the larger to smaller portion is 7/3.

Musical Trio

• A group of musicians includes three drummers, four pianists, and five guitarists.
• Find the number of ways to form a trio with 1 drummer, 1 pianist, and 1 guitarist.
• There are 60 ways to assemble the trio.