Quant Fundamentals Flashcards

Question

multiply whole number by a decimal

Answer 1

Rule 1: When the same constant is added to or subtracted from both sides of an equation, the equality is preserved and the new equation is equivalent to the original equation. Rule 2: When both sides of an equation are multiplied or divided by the same nonzero constant, the equality is preserved and the new equation is equivalent to the original equation. Rule 3: When an expression that occurs in an equation is replaced by an equivalent expression, the equality is preserved and the new equation is equivalent to the original equation.

Answer 2

1. eliminate fractions (if any). multiply by the least common denominator to get rid of fractions. 2. expand parenthesis. use identities as distributive properties. 3. move and combine like-terms together.

Answer 3

In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation. example: 4 x + 3 y = 13 x + 2 y = 2 Express x in the second equation in terms of y as x = 2 −2𝑦. Substitute 2 −2𝑦 for x in the first equation to get 4 (2−2𝑦) +3𝑦 =13. Replace 4 (2−2𝑦) +3𝑦 =13 with (8 - 8y) + 3y = 13 Combine like terms to get 8−5𝑦 =13. Solving for y gives y = -1. now: 4 x + 3 (-1) = 13 x + 2 (-1) = 2 in the second equation: x = 2 - 2 (-1) x = 4

Answer 4

In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other. example: 4 x + 3 y = 13 x + 2 y = 2 multiply the second equation by 4 so they have the same coeffient of x: 4 x + 3 y = 13 4 x + 8 y = 8 if you substract the equations: 4 x - 4 x 3 y - 8 y 13 - 8 you get -5y = 5 y = -1 so x = 2 - 2 (-1) x = 4

Answer 5

A quadratic eqaution: where 𝑎, 𝑏, and 𝑐 are real numbers and a, is not equal to 0𝑎≠0. Quadratic equations have zero, one, or two real solutions. 𝑎𝑥^2+𝑏𝑥+𝑐=0 to solve, you can use the quadratic formula

Answer 6

Some quadratic equations can be solved more quickly by factoring. Example : 2𝑥^2−𝑥−6=0 can be factored as (2𝑥+3)(𝑥−2)=0 product is equal to 0, at least one of the factors must be equal to 0, so either 2x + 3 = 0 or 𝑥−2=0. If 2x + 3 = 0 then 2𝑥=−3 x = -3/2 If 𝑥−2=0, then x = 2 Thus the solutions are negative, −3/2 and 2.

Answer 7

To solve an inequality means to find the set of all values of the variable that make the inequality true. For example, the inequality 4𝑥+1≤ 7 is a linear inequality in one variable, which states that 4𝑥+1 is less than or equal to 7. The procedure used to solve a linear inequality is to simplify the inequality by isolating the variable on one side of the inequality, using the following two rules. Rule 1: When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved. Rule 2: When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original. example: 4𝑥+1≤ 7 substract 1 from both sides 4𝑥 ≤ 6 𝑥 ≤ 6/ 4 𝑥 ≤ 1.5 4(1.5)+1 =7 so x ≤ 7

Answer 8

A function is like a machine: you put in x, do some math, and get an output. Example: f(x)=3x+5, so if x =2 f(2)=11. The domain is the set of numbers we can use. Some functions exclude numbers, like when we cannot divide by 0. The absolute value function gives the distance from zero, so h(x)=∣x∣ means h(−3)=3.

Answer 9

Average= Number of values / Sum of all values

Answer 10

rate and speed are the same.

Answer 11

𝑉=𝑃(1+𝑟𝑡/100) V = final value of the investment P = initial deposit (principal) r = annual interest rate (as a percentage) t = time in years

Answer 12

Unlike simple interest, compound interest keeps growing because you earn interest on top of your interest. 𝑉=𝑃(1+𝑟/100)^𝑡

Answer 13

If interest is compounded quarterly, monthly, or daily, we use this formula: V = P (1+ r /100*n) ^n*t Where n is the number of times per year interest is compounded: Quarterly: n= 4 Monthly: n=12 Daily: n=365

Answer 14

Data can be organized using tables, graphical methods, and numerical methods. A variable represents a characteristic that varies within a population and can be: Quantitative (numerical): e.g., height, age Categorical (nonnumerical): e.g., eye color, political preference. The distribution of a variable describes how frequently different values occur. Frequency: The count of a specific value in the dataset. Relative Frequency: The proportion of a value in relation to the total dataset (expressed as a percentage, fraction, or decimal). Frequency Distributions and Relative Frequency Distributions use tables or graphs to summarize data effectively.

Answer 15

Tables help organize and present data clearly. They are often used for frequency distributions (showing how often values occur) and relative frequency distributions (showing the proportion of each value). A frequency distribution table lists categories or numerical values in one column and their frequencies in another. A relative frequency table follows the same format but shows proportions (percentages, fractions, or decimals) instead of counts. If there are many unique values, data can be grouped into ranges to simplify the table. 1. Understand the Structure of Tables Rows and columns organize data clearly—always check labels to understand what’s being measured. Identify categories (qualitative) vs. numerical values (quantitative). 2. Frequency vs. Relative Frequency Frequency = The number of times a value appears. Relative Frequency = Frequency ÷ Total, expressed as a fraction, decimal, or percentage. Make sure relative frequencies sum to 1 (or 100%). 3. Recognize Grouped Data If there are too many unique values, data is often grouped into ranges. Pay attention to range boundaries (e.g., "71-80" includes both 71 and 80).

Answer 16

Mean (Average) Mean= ∑x / n Sum up all the numbers (∑x) Divide by the total number of numbers (n)

Answer 17

Median Step 1: Arrange numbers in order from smallest to largest. Step 2: If there’s an odd number of values: Median = Middle value If there’s an even number of values: Median = (Middle Value 1+Middle Value 2) /2 Example (odd numbers): Numbers: 4, 4, 6, 7, 10 → Middle number is 6 median = 6 Example (even numbers): Numbers: 1, 2, 3, 4 → Middle numbers are 2 and 3 Median = (2+3) / 2 =2.5

Answer 18

The number that appears most often If one number appears the most → it's the mode If multiple numbers appear the most → multiple modes If no number repeats → no mode Example: Numbers: 1, 2, 3, 3, 4, 4, 4, 5 → Mode = 4 (because it appears the most

Answer 19

Quartiles (Splitting into 4 Parts) Think of quartiles as cutting the list into 4 equal chunks: Q1 (First Quartile): The number that splits off the first 25% of the data. Q2 (Second Quartile or Median): The middle of the whole list (50% mark). Q3 (Third Quartile): The number that splits off 75% of the data. So if we lined up 16 numbers in order, we’d: Find the median (Q2)—the middle of the list. Take the first half of the numbers and find the middle of that → That’s Q1. Take the second half of the numbers and find the middle of that → That’s Q3. For example, if we have this list: 2, 4, 4, 5, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9 Q2 (Median) = 7 (middle number) Q1 = 6 (middle of first half) Q3 = 8.5 (middle of second half)

Answer 20

Now, if we want even smaller sections, we use percentiles, which break the list into 100 tiny pieces. Q1 is the 25th percentile (25% of data falls below it). Q2 (median) is the 50th percentile (halfway point). Q3 is the 75th percentile (75% of data falls below it). Percentiles are useful when dealing with big lists, like test scores or income levels.

Answer 21

Range – The difference between the biggest and smallest number. Range=Maximum Value−Minimum Value

Answer 22

Interquartile Range (IQR) – The middle 50% of your data. This ignores the extreme numbers (outliers) and focuses on the "core" data. Example: If your numbers are 2, 4, 5, 7, 8, 9, the middle half (quartiles) might go from 4 to 8, so IQR = 8 - 4 = 4. formula: IQR= Q3 −Q1 Where: Q 1 Q 1 Q1 (First Quartile) = 25th percentile (middle of the lower half) Q3 (Third Quartile) = 75th percentile (middle of the upper half)

Answer 23

Standard Deviation (SD) – Measures how far each number is from the average. If numbers are super close to the average, SD is small. If they’re spread out, SD is big. Formula: σ = √ ∑(xi −xˉ)^2 / n Where: xi = each data value xˉ = mean (average) of the data n = number of data points Steps: 1. Find the mean: ∑(xi / n 2. Subtract the mean from each data point 3. Square each result 4. Find the average of those squared differences 5. Take the square root of that average Example for numbers 0, 7, 8, 10, 10: Mean = 0+7+8+10+10 =7 Squared differences: (7−0) 2, (7−7)2 ,(7−8) 2, (7−10)2 ,(7−10) 2→ 4, 9,0,1, 9, 9 Average = 49+0+1+9+9 / 5 =13.6 Standard deviation = √ 13.6 ≈ 3.7

Answer 24

Z= (x − xˉ) / σ Where: x = the data value xˉ = mean σ = standard deviation

Answer 25

A set is a collection of objects (called elements or members) that share a common property. Example: The set of even digits: {0, 2, 4, 6, 8}. Types of Sets Finite Set: A set with a countable number of elements. Example: {1, 2, 3, 4}. Infinite Set: A set with an uncountable number of elements. Example: The set of all integers. Empty Set (∅): A set with no elements. Example: The set of all real numbers greater than 5 and less than 5. Nonempty Set: Any set that has at least one element. Subset (⊆): A set A is a subset of set B if all elements of A are also in B. Example: {2, 8} ⊆ {0, 2, 4, 6, 8}. Universal Set (U): The set containing all elements under discussion. Notation Number of Elements in a Set: The number of elements in a set S is denoted as ∣S∣ Example: If S={6.2,−9,π,0.01,0}, then ∣S∣=5. For the empty set: ∣∅∣=0.

Answer 26

A list is a collection of objects in a specific order, where repetitions matter. Example: The lists (1, 2, 3, 2) and (1, 2, 2, 3) are different. a list and a set are different because on lists orders do matter and repetitions are counted.

Answer 27

The intersection of two sets S and T is the set of elements that are in both S and T. Formula: S∩T={x∣x∈S and x∈T} Example: If S={1,2,3} and T={2,3,4}, then S∩T={2,3}.

Answer 28

The union of two sets S and T is the set of elements that are in either S, T, or both. Formula: S∪T={x∣x∈S or x∈T} Example: If S={1,2,3} and T={2,3,4}, then S∪T={1,2,3,4}.

Answer 29

Two sets are disjoint if they have no elements in common. Formula: S∩T=∅. Example: If A={1,2,3} and B={4,5,6}, then A∩B=∅.

Answer 30

A Venn diagram is a visual representation of sets and their relationships using overlapping circles. Each circle represents a set. Overlapping regions represent intersections. The universal set (U) is often represented by a rectangle containing all sets.

Answer 31

A formula used to count elements in the union of two finite sets while avoiding double-counting elements in their intersection. Formula For two sets A and B: ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣ For two disjoint sets B and C: ∣B∪C∣=∣B∣+∣C∣ (B∩C=∅, so no need to subtract the intersection). Example If: ∣A∣=30 ∣B∣=25 ∣A∩B∣=10 Then: ∣A∪B∣=30+25−10=45

Answer 32

The Multiplication Principle states that if one event can occur in m ways and a second event can occur in n ways, then the total number of ways both events can occur together is: Total Ways=m×n General Formula If a sequence of k independent events occurs in n1, n2, .... nK ways respectively, then the total number of ways all events can happen is: n1* n2* ...*nK Example A restaurant offers 4 appetizers, 3 main courses, and 5 desserts. The number of different meals (one appetizer, one main course, and one dessert) is: 4×3×5=60

Answer 33

A permutation is just a fancy word for arranging things in a specific order. Order matters in permutations. Formula: P(n,k)= n! / (n−k)! n = total objects k = number of objects you’re picking Example: If you want to pick 5 digits from 7 and arrange them: P(7,5)=7! / (7−5)! = 7! / 2! = 7×6×5×4×3×2×1 / 2×1 Cancel out 2 × 1 (because it appears in both the top and bottom): 7×6×5×4×3=2,520 So, there are 2,520 ways to make a 5-digit number using 5 unique digits from 1 to 7.

Answer 34

A factorial is when you multiply a number by all the numbers before it. It’s written as n! (read as "n factorial"). formula: n!=n×(n−1)×(n−2)×...×3×2×1 Examples: 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24 5! = 5 × 4 × 3 × 2 × 1 = 120 0! is always 1 (just a rule to make math easier). Factorials help us quickly count how many ways we can arrange things.

Answer 35

Combinations are used when you want to choose items from a set, but the order doesn’t matter. Formula for Combinations (n choose k): C(n,k)= n! / k!(n−k)! Where: n = total number of items k = number of items you're choosing ! = factorial Example: How many ways can you pick 3 students from a group of 5? C(5,3)= 5! / 3!(5−3)! = 5! / 3!2 = 5×4×3! / 3!×2×1 = 5×4 / 2× =10 So there are 10 ways to choose 3 students from 5.

Answer 36

Definition: Probability is a numerical way to describe uncertainty. General Probability Rules Rule 1 (Certain & Impossible Events): P(certain event)=1 P(impossible event)=0 Rule 2 (Complement Rule): The probability that an event does not occur: P(not E)=1−P(E) Rule 3 (Sum of All Probabilities): The sum of the probabilities of all possible outcomes is 1.

Answer 37

If all outcomes are equally likely, the probability of an event E is: P(E)= Number of outcomes in E/ Total number of outcomes in sample space Example 1: Probability of rolling a 4: P (4) = 1/6

Answer 38

Two events cannot happen at the same time. Example: Rolling an odd number and rolling an even number. Formula: P(E or F)=P(E)+P(F)

Answer 39

Two events do not influence each other. Example: Rolling a die twice → The result of the first roll does not affect the second roll. Formula: P(E and F)=P(E)×P(F)

Answer 40

How numerical data is spread out or organized.

Answer 41

The proportion of times a value appears compared to the total data set. formula : frequency of value / total number or data points

Answer 42

Measures how spread out the data is around the mean (average). d= √∑(xi −m)^2 / n Measures how spread out the data is around the mean. Standard Deviation Ranges 1 SD from mean: m±d → Includes ~68% of data 2 SD from mean: m±2d → Includes ~95% of data 3 SD from mean: m±3d → Includes ~99.7% of data

Answer 43

Total Area=1 The total area under a probability distribution (or histogram bars) always equals 1 (or 100% of data).

Answer 44

is just a way to represent uncertain outcomes with numbers. Instead of saying "something random happens," we assign a number to each possible outcome.

Answer 45

is the average outcome you'd expect if you repeated the experiment many times. This is used when you have probabilities associated with different values. The formula is: E(X)=∑X⋅P(X) where: X: is each possible value of the random variable P(X): is the probability of that value

Answer 46

is a bell-shaped curve that represents naturally occurring data. The bell-shaped curve appears when you graph the normal distribution on a coordinate plane.

Answer 47

Mean (m): The middle of the data, also called the average. It’s the highest point on the bell curve. Standard Deviation (d): A measure of how spread out the data is. A small d means the data is tightly packed near the mean, while a large d means the data is more spread out. The bell curve is symmetric, meaning the left and right sides look the same.

Answer 48

1. Mean = Median = Mode → The most common value is also the average, so everything is centered. 2. Symmetry → The left and right sides of the curve are mirror images. 3. most values (around 68%) aren't too far from the average. One standard deviation (σ) tells us how spread out the data is. If data is normally distributed, about 68% of values fall between: Mean−σ to Mean+σ example: If test scores have a mean of 75 and a standard deviation of 5, About 68% of students scored between 70 and 80 (75 ± 5). 4. If we go a bit farther from the mean (2 standard deviations), we cover almost all values (95%). If data is normally distributed, about 95% of values fall between: Mean−2σ to Mean+2σ Example: Using the same test score example (mean = 75, standard deviation = 5), About 95% of students scored between 65 and 85 (75 ± 10).

Answer 49

The total area under the curve represents 100% of the data. If you randomly pick a person, the chance of them falling in a certain range is the area under that section of the curve. Example: The probability of someone being taller than the average is 50% (since the curve is symmetrical). The probability of someone having an IQ between 85 and 115 is 68% (because that’s within 1 standard deviation).

Answer 50

This is just a special normal distribution where the mean is 0 and the standard deviation is 1. Any data point can be converted into this form using the formula: Z= (X−m) / d Where: X = the data value m = the mean d = the standard deviation Example: If your test score is 85, the average is 70, and the standard deviation is 10: Z = (85−70) / 10 = 15/10 =1.5

Answer 51

(x + a)(x − a) = x² − a²

Answer 52

(a + b)² = a² + 2ab + b² ## Footnote Also, (a − b)² = a² − 2ab + b²

Answer 53

a³ − b³ = (a − b)(a² + ab + b²)

Answer 54

a³ + b³ = (a + b)(a² − ab + b²)

Answer 55

ab + ac = a(b + c) ## Footnote Also, a²b − ab = ab(a − 1)

Answer 56

(x + a)(x + b) = x² + (a + b)x + ab

Answer 57

Find two numbers that multiply to c and add to b. ## Footnote Example: x² + 5x + 6 = (x + 2)(x + 3)

Answer 58

If x² = a, then x = ±√a

Answer 59

x = [−b ± √(b² − 4ac)] / 2a