Essential Quant Skills Flashcards by Alvaro Nicolau

What is a Proper, Improper, and Mixed Fraction?

Proper Fraction: 1/7, 3/4, 5/9, 4/41
Improper Fraction: 6/2, 18/5, 31/7, 4/3
Mixed Fraction: An Integer + Proper/Improper Fraction

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Convert an improper fraction into a mixed number. Ex 7/2 and 18/3

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Convert a mixed fraction into an improper fraction. Ex 3 1/2 and 4 2/3

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What is the Least Common Denominator (LCD)?

The LCD of two or more fractions is the smallest Non-Zero whole number that is divisible by each of the denominators.

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How do you determine the LCN of a group of fractions?

1.- Prime Factorization of denominators in each fraction

2.- Use the prime that repeats the most in each number

3.- Square prime if it repeats in that same number

4.- Multiply the Primes to find the LCN

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How do you determine the LCD of a group of fractions?

1.- Prime Factorization of denominators in each fraction

2.- Use the prime that repeats the most in each number

3.- Square prime if it repeats in that same number

4.- Multiply the Primes to find the LCD

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Problem: Add 7/8 and 5/3 using LCD

61/24

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Problem: Substract 7/3 and 9/5 using LCD

8/15

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How can you calculate if a pair of fractions are equivalent?

Is 7/56 = 10/80?

7*80=560

56*10=560

Fractions are equivalent

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Simplifying Fractions - What is the Top-and-bottom method?

Solve 7/77 * 36/72 * 15/45 using the T&B method

7/77 Divide by 7 = 1/11

36/72 Divide by 36 = 1/2

15/45 Divide by 5 = 3/9 –> 1/3

Multiply the 3 results, and you get = 1/66

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Simplifying Fractions - What is the Cross Simplification method? And what are the unique conditions for using this method?

Solve 35/64 * 56/45 using the Cross Simplification method

In Cross Simplification, we simplify the fractions by taking the numerator of one fraction and the denominator of another fraction and dividing both numbers by the same Value.

Unique conditions:

1.- Cross Simplifications can be performed on fractions that are not adjacent. In other words, they don’t need to be next to each other.

2.- This method only works with multiplication and division problems.

Problem: 35 & 45 get divided by 5
64 & 56 get divided by 8

Resulting in 7/8 * 7/9 = 49/72

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What is a reciprocal and how do you get it? What is a Unique condition?

Give me the reciprocals of 5, 8, -2, -8, 1/5, 2/5, -2/3, -9/4,

A reciprocal is the inverse of a value or number.
To get the reciprocal of a non-zero number, divide 1 by that number or put 1 over that number.
If it’s a negative number or fractions, the negative sign stays in the numerator.

1/5, 1/8, -1/2, -1/8, 5/1, 5/2, -3/2, -4/9

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Which numbers/values don’t have a reciprocal? And a special rule associated with the multiplications of reciprocals.

Solve: in which of the following number pairs are the two numbers reciprocal of one another:

1) -19/171 & -171/19

2) 9^9 divided by 1/9 and 1/9^10

3) -2 & 1/2

1.- The reciprocal of 1 is 1, and the reciprocal of -1 is -1.

2.- Zero is the only number that doesn’t have a reciprocal. 0 is 0

3.- The product (multiplication) of a number and its reciprocal is 1

Solve:

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What is a Complex Fraction?

Complex Fractions are fractions whose numerator, denominator, or both are also fractions. See image

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Which is the first method you can employ to simplify correctly a Complex Fraction?

Solve the image.

Write the numerator and denominator as single fractions and the divide.

6/17

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Which is the second method you can employ to simplify correctly a Complex Fraction?

Solve the image using the second method.

Multiply both the numerator and the denominator of the complex fraction by the LCD, and then Simplify

6/17

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What is the Bow Tie Method?

Solve: what is larger = 7/9 or 6/8?

It’s a method used between two positive fractions to know which one is larger.

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In a set of positive fractions, if all of them share the same denominator, which fraction is the largest?

Solve the following example: 7/12, 1/3, 3/4, X, 1/5

The set above contains 5 unique fractions. If X is the second greatest fraction in the set, which of the following could be X?

1.- 32/60

2.- 35/60

3.- 38/60

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The fraction with the largest numerator is the larger fraction.

We need to use LCD to solve the problem, and usually, that is the strategy. LCD is 60 for this problem.

Result 38/60

How do you distribute the property of a fraction with the same denominator but different numerators?

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Also, 495/9 = 450/9 + 45/9 = 50 + 5 = 55

What is the Least Common Numerator (LCN), and when should we use it?

Solve the example.

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The LCN is the same as the LCD but for the numerator, even though its only use is for comparing the size of fractions because we can not use it to add, subtract, multiply, or divide since we would get a wrong answer.
If the numerator of two or more fractions is the same, then the smaller the denominator, the bigger the fraction, and the bigger the denominator the smaller the fraction.
Answer 1 & 3

What happens to a fraction between 0-1 when we add or subtract a constant number to the numerator and denominator? Provide an example for both cases.

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If we add a positive number, the fraction gets BIGGER.

For example, we add 6 in the numerator and denominator to 3/4, equalling = 9/10. Comparing both fractions 9/10 (.9999) is bigger than 3/4 (.75555)

If we add a negative number, the fraction gets SMALLER.

For Example, we subtract -2 in the numerator and denominator to 3/4, equalling = 1/2. Comparing both fractions 3/4 (.75) is bigger than 1/2 (.5)

What happens to a fraction greater than 1 when we add or subtract a constant number from the numerator and denominator? Provide an example for both cases.

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If we add a positive number, the fraction gets SMALLER.

For example, we add 4 in the numerator and denominator to 10/3. Equalling = 14/7. Comparing both fractions 14/7 is SMALLER than the original fraction 10/3.

If we add a negative number, the fraction gets BIGGER.

For example, we subtract -1 in the numerator and denominator to 10/3. Equalling = 9/2. Comparing both fractions 9/2 is BIGGER than the original fraction 10/3.

Name the place values of a number and give and example of each.

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How do you round up and down a positive integer?

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When rounding a positive integer to a specific place value, we have to look at the DIGIT IMMEDIATELY TO THE RIGHT of that place value.
If that digit to the right is 0, 1, 2, 3, or 4, we ROUND DOWN by keeping the digit in the original place value the same, and we change all digits to the right of that place value to zero.
If the digit to the right is 5, 6, 7, 8, or 9, then we round up by adding 1 to the digit in the original place value, and we change all digits to the right of that place to zero.

Solve the following exercises: - What is 123 and 176, and 195 rounded to the nearest ten? - What is 153 and 2433 rounded to the nearest hundred? - What is 2,764 and 7,482 rounded to the nearest thousand? - What is 0.343 and 0.144 rounded to the nearest tenth? - What is 0.179 and 0.628, and 0.397ounded to the nearest hundredth?

- 120 & 180 & 200 - 200 & 2400 - 3,000 & 7,000 - 0.3 & 0.1 - 0.18 & 0.63 & 0.4

What does the percentage mean? And how do you turn a percentage into a fraction? Solve the following exercises: - What are 2%, 30%,10%, 1% in fraction? - What is X% and Y% in a fraction? What is 5% + 7% in a fraction? - What is (X%+Y%)*Z%

- Percentage means divide by 100. X% =X/100 - 2/100, 30/100, 10/100, 1/100 - X/100 + Y/100 = 5/100 + 7/100 = 12/100 = 6/50 = 3/25 - XZ+YZ/10,000

How do you convert a fraction into a percentage? - Convert the following fractions into percentages: - 7/100 - 1/4 - 3/2 - 1/3

- To convert a fraction into a percentage, multiply the fraction by 100 and attach a percentage sign. - (7/100) *100 = 7% - (1/4) * 100 = 100/4 = 25% - (3/2) *100 = 300/2= 150% - (1/3)*100 = 100/3 = 33.33%

How do you convert a decimal into a percentage? - Convert the following numbers into percentages: - 0.03 - 0.85 - 2 - 0.004

To convert a decimal or an integer to a percentage, move the decimal point two places to the right and attach a percentage sign. - 3% - 85% - 200% - 0.4%

How do you convert a percentage into a decimal? - Convert the following percentages into decimals - 0.7% - 36% - 298%

To convert a percentage to a decimal, drop the percentage sign and move the decimal point two places to the left. - 0.007 - 0.36 - 2.98

How do you convert a decimal into a fraction? Convert the following examples: - 0.3 - 1.47 - 0.679 - 0.0056

Depending on the number of decimals a decimal has, the numerator is the number without the decimal point, and the denominator is 1 followed by N zeros. - 3/10 - 147/100 - 679/1,000 - 56/10,000

- How do powers affect fractions with parenthesis and without parenthesis? Provide an example for both cases. - What happens when you take the square root of a number between 0 and 1 (proper fraction), and when you square the same fraction? Provide an example with X= 1/16

- There are two ways to get the square root of a fraction. 1.- If the fraction has a parenthesis next to an exponent, then the numerator and denominator are affected by the exponent. For example (2/5)^2 = 4/25 2.- If the fraction does NOT have a parenthesis next to an exponent, then it only affects the numerator and NOT the denominator. For example 2/5^2 = 4/5 - 0 < 1/16 < 1 and 1/256 < 1/16 < 1/4

When presented with a large number raised to the second power, how do we know which unit digit number the result is going to end in? Solve the following examples, what unit digit is the result going to end: - 309,579^2 - 5,678,244^2 - 504.13^2

When you raise a number to the second power (perfect square), then we can square just the units digit of the original number to determine the units digit of the full result. - 81, the result is going to end in 1 - 16, the result is going to end in 6 - 9, the result is going to end in 9

If a number is a perfect square which numbers can and cannot end in?

- It can end in 0,1,4,5,6,9 -It cannot en in 2,3,7,8

Which is a good strategy for comparing the size of two negative fractions? Which of the following is the smallest one? -5/6 -3/4 -1/2 -6/7 -1/9

When comparing the size of two negative fractions: 1.- We should first assume that they are positive (that they don't have a negative sign). 2.- Then, once we have determined which number is the largest (without the negative sign), that "largest number" is actually the SMALLEST NEGATIVE NUMBER.

What is the complete order operations? Which ones are it's time saving variations?

PEMDAS - Plus P - Parentheses + Absolute Value + Square Root E - Exponents M - Multiplication D - Division A - Addition S - Substracion * ALWAYS solve left from left to right. Time-saving variations 1.- If the operations involved are only addition and subtraction or the only ones remaining (No parentheses, multiplication, exponent, or division) we don't have to add or subtract from left to right. 2.- Simplifying Each term - Terms that are separated by a plus or a minus sign, that are outside of a parenthesis we CAN divide them into group terms.

Solve the following exercises:

8/3

Solve the following exercises:

23.5

What is the distributive property? - What are three types of formulas?

The distributive property states that multiplying a number by a sum of two numbers gives the same result as multiplying this number by each addend and then adding the products together. - A* (B+C) = A*B + A*C - (B+C)*A = B*A + C*A - A (B+C+D) = A*B + A*C + A*D * The same formula applies for subtracting values

Solve the following examples: 1. 43(16) + 32(16) + 21(16) + 10(16) - 6(16)

1. 1,600 2. 5/6

How can we rearrange numbers with addition and subtraction to perform easier mathematical operations? Solve the following operations: - Add 999 and 578 - 692 + 996 - 9,992 + 6,889 - 1,008 + 995 + 1,007 + 993 - 30(779) + 15(779) + 15(779) + 20(779) + 15(779) + 4(779) + 798 1,000 - 857 1,005 - 667 10,011 - 7,677 1,000,000,000,000 - 888,888,888,888

- By adding or subtracting from the original number. - 1,000 - 1 + 578 = 1,577 - 1,000 + 692 - 4 = 1,688 - 10,000 - 8 + 6,889 = 16,881 - 1,000 + 8 + 1,000 - 5 + 2,000 = 4,000 + 3 = 4,003 - 99(779) = (100 (779)) - 1 = 77,900 - 1 = 77,899 999 - 857 + 1 = 142 + 1 =143 999 - 667 + 6 = 332 + 6 = 338 9,999 - 7,677 + 12 = 2,322 + 12 =2,334 999,999,999,999 - 888,888,888,888 + 1 = 111,111,111,111 + 1

What is a factorial, and what does it represent? Provide an example Which are the two special cases in factorials? Solve the following: If N = 5! and M =4!, what is N-M? If P! - Q! = 0, what has to be P and Q?

- A factorial is a positive integer followed by an exclamation mark; imagine 8! and it represents the product of all integers from 1 to n (or in this case, 8) - 8! = 8*7*6*5*4*3*2*1 = 40,320 - 0! = 1 and 1! = 1 - 5! = 5*4*3*2*1=120 and 4! = 24, so 120 - 24 = 96 - 0 OR 1

How do you simplify a factorial? Give an example for 5! and 8! Solve the following: - 10!/(8! * 3!) - If X = 10! and Y = 56 x 6!, what is X/Y? If R = 10! * 3! * 2! / 9! * 3 * 2 and S = 8! * 7! / 5! * 6! * 3! * 7. What is R/S?

5! = 5 * 4! or 5 * 4 * 3! or 5 * 4 * 3 * 2! or 5 * 4 * 3 * 2 * 1! 8! = 8 * 7 * 6! or 8 * 7 * 6 * 5! or 8 * 7 * 6 * 5 * 4 *3!.... Exercises: - 15 - I & III - 90 - 5/14

Usually, when do we have to factor factorials? Solve the following examples: 11! + 12! = ? 9! - 8! - 7! = ? 20! + 19! + 18! 81! - 80! + 79! / 79! 101!-100! / 99! - 98!

- Usually, we factor factorials when we have to ADD or SUBSTRACT them from each other. Also, we need to use the skill of shortening factorials. By shortening factorials, we can more easily spot the greatest term that can be factored from each factorial expression. - 11! + 12 * 11! = 11! (12 + 1) =11! (13) 9! = 9 * 8 * 7! and 8! = 8 * 7! and 7! 7! (9 * 8 - 8 - 1) = 7! (72 + 9) = 7! (63) - 18! (400) - 80^2 + 1 or 6401 - 10,000 Aprox

Essential Quant Skills Flashcards

(43 cards)