Mathematical Concepts and Word Problems Practice

A set of vocabulary-style flashcards covering mathematical logic, algebraic equations, and word problems from the lecture notes.

Tank Capacity Calculation

Given a fuel tank is $\frac{4}{7}$ full and requires $32\,gallons$ more to become $\frac{3}{5}$ full, the total capacity is calculated using the equation $\frac{3x}{5} - \frac{4x}{7} = 32$, resulting in $140\,gallons$.

Fraction Comparison Rules

Among fractions with equal denominators, the fraction with the larger numerator is the greater fraction; for instance, $\frac{5}{6}$ (equal to $\frac{55}{66}$) is greater than $\frac{8}{11}$ (equal to $\frac{48}{66}$).

Decimal to Simple Fraction Conversion

The repeating decimal $0.47$ is converted to a general fraction by subtracting the non-repeating part from the total number and dividing by a denominator of $90$, resulting in $\frac{43}{90}$.

Fractional share valuation

If a son sells $\frac{2}{3}$ of his $\frac{3}{5}$ share of a property for $tk.\,1,00,000$, the total value of the original property is $2,50,000\,tk$.

Remaining Reading Pages Calculation

If $96$ pages of a book have been read and $\frac{5}{13}$ of the book is left to read, the total page count is found by setting the read portion $\frac{8}{13}$ equal to $96$, resulting in $156$ pages.

Box Method for Polls

A method to determine specific subsets in a population; for example, if $\frac{2}{5}$ of people polled said 'yes' to question 1 and $\frac{1}{3}$ of those said 'yes' to question 2, the portion who did not answer 'yes' to both is $\frac{13}{15}$.

Quadratic Equation Solution Set

For the equation $x^2 - (p + q)x + pq = 0$, the solution set is $\{p, q\}$ because the expression factors into $(x - p)(x - q) = 0$.

Unique Solution Condition for Linear Equations

A system of equations like $3x + 4y = 12$ and $kx + 12y = 30$ does not have a unique solution if the ratios of the coefficients of $x$ and $y$ are equal ($\frac{k}{3} = \frac{12}{4}$), which occurs when $k = 9$.

Greatest Value of a Quadratic in a Range

For the expression $x^2 - 10x + 16$ within the inclusive range $-4$ to $4$, the greatest value achieved is $72$ when $x = -4$.

Linear Equation for Simultaneous Costs

Given $4\,gel-pens + 8\,ball-point\,pens + 1\,marker = 185\,tk$ and $7\,gel-pens + 15\,ball-point\,pens + 1\,marker = 315\,tk$, the total cost of one of each item is $55\,tk$.

Three-Digit Number Property

A 3-digit number where the sum of the first two digits equals the third ($x + y = z$) and reversing the digits increases the value by $198$ ($100z + 10y + x - (100x + 10y + z) = 198$) has $7$ possible integer solutions.

Cyclic Exponent Identity

If $x = y^a$, $y = z^b$, and $z = x^c$, then the product of the exponents $abc$ must equal $1$.

Sum of Powers with Same Base

The expression $4^x + 4^x + 4^x + 4^x$ simplifies to $4 \times 4^x$, which is equal to $2^2 \times (2^2)^x = 2^{2x+2}$.

Trailing Zeros in Large Products

The number of consecutive zeros to the left of the decimal point for the integer $(125)^{14} \times (48)^8$ is $32$, derived from the shared power of $10$ in its prime factorization ($5^{42} \times 2^{32} \times 3^8$).

Remaining Fraction after Sequential Subtraction

If three people sequentially take $\frac{3}{5}$ of the remaining marbles in a box, the final fraction of marbles left is calculated as $(1 - \frac{3}{5})^3 = (\frac{2}{5})^3 = \frac{8}{125}$.

Attendance Ratio in Club Members

If $\frac{2}{3}$ of male and female members are absent and $\frac{1}{3}$ of those present are male, the ratio of absent males to absent females is $7:5$.

Integer Constraint in Linear Equations

In the equation $7x - 4y = 20$ where $x$ and $y$ are integers, $x$ must be a multiple of $4$ for the term $7x$ to be divisible by $4$ (since $20$ and $4y$ are both divisible by $4$), making $8$ a valid solution for $x$.

Arithmetic Series in Word Problems

Given seven balls where each costs $n\,taka$ more than the next smaller one, the biggest costs $46\,tk$, and the sum is $196\,tk$, the common difference $n$ is $6$.

Merit List Total Calculation

If a student is $15th$ from the top and $30th$ from the bottom among those who passed, the total class size (including $6$ non-participants and $10$ failures) is $14 + 1 + 29 + 6 + 10 = 60$.

Proportional Marble Distribution

If three friends have marbles based on $x = 6y = 3z$, for all to have an equal count, the person with $x$ marbles must give away $\frac{1}{2}$ of their total amount.