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Pre-Algebra Notes
KK Pre-Algebra Notes
| Heading | Example |
|---|---|
| Converting Mixed Numbers to Improper Fractions: 2 3/7 | 2 3/7 = 2 + 3/7 = 7+2/7+3/7 = 7 x 2 + 3/ 7 = 14 + 3/7 = 17/7 |
| Converting Improper Fractions to Mixed Numbers: 19/6 | 19/6 = 18 + 1/6 = (6 x 3) + 1/6 = 6 x 3/6 + 1/6 = 3 + 1/6 = 3 1/6 |
| Negative Fractions: -2 3/7 | -2 3/7 = - (2 + 3/7) = - (7 x 2/7 + 3/7) = - [(7 x 2) + 3/7] = -17/7 |
| Negative Fractions: - 19/6 | - 19/6 = - (18 + 1/6) = [(6 x 3) + 1/6] = - (6 x 3/6 + 1/6) = - (3 + 1/6) = - 3 1/6 |
| Adding and Subtracting Mixed Numbers: 6 3/7 + 2 1/4 | (6+2) + (3/7 + 1/4) = (6+2) + [3/7 (4/4) + 1/4(7/7)] = (6+2) + (12/28 + 7/28) = 8 + 19/28 = 8 19/28 |
| Converting Mixed Numbers to Improper Fraction: 8 19/28 | 8 19/28 = (28 x 8) + 19 /28 8 19/28 = 224 + 19 /28 8 19/28 = 243/28 |
| Adding and Subtracting Mixed Numbers: 6 3/7 - 2 1/4 | (6 - 2) + (3/7 - 1/4) = (6 - 2) + [3/7 (4/4) - 1/4 (7/7)] = (6 - 2) + (12/28 - 7/28) = 4 + 5/28 = 4 5/28 |
| Converting Mixed Numbers to Improper Fractions: 4 5/28 | 4 5/28 = (28 x 4) + 5 /28 4 5/28 = 112 + 5 /28 4 5/28 = 117/28 |
| Multiplying and Dividing Mixed Numbers Find the product: 2 2/3 x 5 3/7 | [(3 x 2) + 2 /3] x [(7 x 5) + 3 /7] (6 + 2 /3) x (35 x 3 /7) = 8/3 x 38/7 8 x 38 / 3 x 7 = 304/ 21 = 14 10/21 |
| Multiplying and Dividing Mixed Numbers Find the quotient: 4 1/6 (divided by) 3 1/3 | [(6 x 4) + 1 /6] (divided by) [(3 x 3) + 1 /3] (24 + 1 /6) (divided by) (9 + 1 /3) = 25/6 (divided by) 10/3 25/5 x 3/10 = 75/60 = 75 (divided by) 15 /60 (divided by) 15 = 5/4 = 1 1/4 |
| Which is greater? 5/7 or 2/7 | 5/7 = 2/7 < 5/7 |
| Which is greater? 5/8 or 2/3 | 5/8 = 5/8 (3/3) = 5 x 3 / 8 x 3 = 15/24 2/3 = 2/3 (8/8) = 2 x 8 / 3 x 8 = 16/24 = 2/3 > 5/8 |
| What is two - fifths of the way from 3 and 8 | 8 - 3 = 5 (divided by) 5 = 5/5 = 1 1 x 2 = 2 = 3 + 2 = 5 = 5 is two-fifths(2/5) of a unit away from 3 and 8 |
| What is two-thirds of 6 | 2/3 x 6 = 2/3 x 6/1 = 2 x 6 / 3 x 1 = 12/3 = 4 So two-thirds of 6, is 4(or 2/3 of six is 4) |
| Find a number that's 1/2(half) of the way from 1/7 to 6/11 | 6/11 - 1/7 = 6/11 (7/7) - 1/7 (11/11) = 42/77 - 11/77 = 31/77 = 31/77 x 1/2 = 31/154 = 1/7 + 31/154 = 1/7 (22/22) + 31/154 = 22/154 + 31/154 = 53/154 = 53/154 is the number that's 1/2 of the way from 1/7 to 6/11 |
| Find the sum of the mixed measures: 3 yards, 2 feet, 4 inches 6 yards, 2 feet, 8 inches | (3 + 6)yards, (2 + 2)feet, (4 + 8)inches = 9 inches, 4 feet, 12 inches 9 yards, 4 feet, 12 inches = 9 yards, 4 feet, 1 foot = 9 yards, (4 + 1) feet = 9 yards, 5 feet 9 yards, 3 feet + 2 feet = 9 yards, 1 yard + 2 feet = (9 + 1)yards, 2 feet = 10... |
| To be continued with: Find the sum of the mixed measures: 3 yards, 2 feet, 4 inches 6 yards, 2 feet, 8 inches | ...10 yards, 2 feet This is the sum of 3 yards, 2 feet, 4 inches and 6 yards, 2 feet, 8 inches |
| Express 45 minutes as a fraction and a decimal | Fraction: 3/4 hours Decimal: 0.75 hours |
| Place Value: In which place is the 7 located in: 32. 18476 | 7 is in the "ten-thousandths" place |
| Find the place values of all the numbers in 2,635.487 | 2 = thousands, 6 = hundreds, 3 = tens, 5 = ones(units) 4 = tenths, 8 = hundredths, 7 = thousandths |
| Find the expanded notation of 500 | 5(100) + 0(10) + 0(1) [You can cancel out the zero's multiplication] = 500 |
| Find the expanded notation of 1,281 | 1(1000) + 2(100) + 8(10) + 1(1) = 1,281 |
| Find the expanded notation of .55 | 5/10 + 5/100 = .55 |
| Find the sum and difference: 13.16 + 8.74 13.16 - 8.74 | 13.16 13.16 + 8.74 - 8.74 --------- --------- 21.90 4.42 |
| Find the product of: 13.1 x 8.74 | 13.1 x8.74 --------- 524 9170 +104800 ------------- 114494 = 13.1 x 8.74 = 114.494 |
| Find the quotient: 13.1 ÷ 8.74 | ______1.4988.....___ 874|1310.00000 - 874 --------- 4360 - 3496 ---------- 8640 - 7866 ----------- 7740 ..... = 13.1 ÷ 8.74 ≈ 1.4988 |
| Ratio and Proportion: Solve for the unknown: 4/5 = 2x/40 | 40 (4/5) = 40 (2x/40) = 40 (4/5) = 2x = 160/5 = 2x = 32 = 2x = 32/2 = 2x/2 = 16 = x = x = 16 |
| Solve for the variable: 1/6x = 3/20 | 20(1) = 6x(3) = 20 = 18x = 20/18 = 18x/18 = 20/18 = x 10/9 = x |
| Unit Price: Which jar of peanut butter is a better value: Jar A costs $2.00 for 1 pound and Jar B $1.80 for 12 ounces | $1.80 / 12 ounces = x / 1 pound = $1.80 / 12 ounces = x / 16 oz. = $1.80 / 12 oz. (16 oz.) = x = $1.80 (16/12) = x = $1.80 (16 ÷ 4 / 12 ÷ 4) = x = $1.80 (4/3) = x = $1.80(4) /3 = x = $7.20 /3 = x =$2.40 = x |
| Convert the value from meters to centimeters(Hint: 100 centimeters are in one meter): 3.5 meters | 100 centimeters / 1 meter = 3.5 x 100 centimeters / 1 = 350 centimeters / 1 = 350 centimeters |
| Convert the value from inches to yards(Hint: 12 inches = 1 foot, and 3 feet = 1 yard): 288 inches | 288 inches x 1 foot / 12 inches x 1 yard / 3 feet = 288 x 1 foot/12 x 1 yard/3 = 288 x 1 x 1 / 12 x 3 yards = 288/12x3 yards = 288 ÷ 3 / 12 x (3 ÷ 3) = 96/ 12 x1 = 96/12 yards = 8 yards |
| Exponents: Write 4 x 4 x 4 x 4 x 4 x 4 as an exponent | 4 to the power of 6 |
| Exponents: Write 6 to the power of 3 in expanded form | 6 x 6 x 6 = 216 |
| Exponents: Write the variable x to the power of 3 in expanded form | (x) (x) (x) = x to the power of 3 |
| Adding Exponents: What is 3x^2 + x^2 | 3x^2 + x^2 = (x^2 + x^2 + x^2) + x^2 = x^2 + x^2 + x^2 + x^2 = 4x^2 |
| Subtracting Exponents: What is x^2 - 3x^2 | 3x^2 - x^2 = (x^2 + x^2 + x^2) - x^2 = x^2 + x^2 + x^2 - x^2 = x^2 + x^2 + (x^2 - x^2) = x^2 + x^2 + 0 = x^2 + x^2 = 2x^2 |
| Multiplying Exponents: What is x^4 * x^5 | x^4 * x^5 = (xxxx) * (xxxxx) = (xxxxxxxxx) = x^9 |
| Dividing Exponents: What is x^5 ÷ x^2 | x^5 ÷ x^2 = x * x * x * x * x / x * x = (x * x * x * x * x) ÷ (x * x) / (x * x) ÷ (x * x) = x * x * x / 1 = x^3 |
| The Power Rule for Exponents: (2^2)^4 | 2^2 x 4 = 2^8 = 256 |
| The Power Rule for Negative Exponents: (3^2)^-2 | (3^2)^-2 = 3^2(-2) = 3^-4 = 1/3^4 = 1/81 |
| Quotient Rule for Exponents: x^4/x^3 | x^4/x^3 = x^4-3 = x^1 = x |
| Quotient Rule for Negative Exponents: x^3/x^-2 | x^3/x^-2 = x^3-(-2) = x^3+2 = x^5 |
| Quotient Rule for Exponents: x^5/x^7 | x^5/x^7 = x^5-7 = x^-2 = 1/x^2 |
| What is the √9 | √9 = 3 = 9^1/2 |
| What is the √0 | √0 = 0 = 0^1/2 |
| What is the √x, if x is negative | √x = x^1/2 = But if x is negative, then √x and x^1/2 are undefined |
| What is the √x, if x is positive | √x = x^1/2 = If x is positive, then √x and x^1/2 is defined |
| What is the √-9 | √9 = (3)(3) or (-3)(-3) = Which, in the end, will just equal positive 9 For now, we will only focus on positive roots until algebra |
| Adding Radicals: √3 + 4√3 | √3 + 4√3 = 1√3 + 4√3 = 5√3 |
| Subtracting Radicals: √5 - √3 | √5 - √3 = Since the radicands have different terms, we cannot simplify this expression at all |
| Adding Radicals: √2 + √8 | √2 + √8 = √8 = √(4)(2) = (√4)(√2) = 2√2 = √2 + 2√2 = 1√2 + 2√2 = 3√2 |
| Multiplying Radicals: √3 √2 | √3 √2 = √3 x 2 = √6 |
| Multiplying Radicals: √6 | √6 = √3 x 2 = √3 √2 |
| Multiplying Radicals: √5 √5 | √5 √5 = √5 x 5 = √25 |
| Multiplying Radicals: √25 | √25 = 5 |
| Multiplying Radicals: (4√2) √3 | (4√2) √3 = 4(√2 √3) = 4√2 x 3 = 4√6 |
| Dividing Radicals: √6 ÷ √3 | √6 ÷ √3 = √6 ÷ 3 = √2 |
| Dividing Radicals: √2 | √2 = √6 ÷ 3 = √6 ÷ √3 |
| Dividing Radicals: √5 ÷ √5 | √5 ÷ √5 = √1 = 1 |
| Rationalize the Denominator: 7/√5 | 7/√5 = 7 √5 / √5 √5 = 7 √5 / 5 |
| Radical Expressions: 3√2 + 6√8 - √18 | 3√2 + 6√8 - √18 = 3√2 + 6√4 x 2 - √9 x 2 = 3√2 + 6√4√2 - √9√2 = 3√2 + 6(2)√2 - 3√2 = 3√2 + 12√2 - 3√2 = (3 + 12 - 3)√2 = 12√2 |
| Power of 10: 67 x 1,000 | 67 x 1,000 = 67.000 = 67,000 |
| Power of 10: 4.3 x 100 | 4.3 x 100 = 04.3 = .043 = 0.043 |
| Power of 10: 510.75 x 10^-2 | 510.75 x 10^-2 = 5.1075 |
| Express the Number in Scientific Notation: 0.000000000000000000000000782 | 0.000000000000000000000000782 = 7.82 x 10^25 |
| Multiplying Scientific Notation: (3.4 x 10^-6) (2.14^13) | (3.4 x 10^-6) (2.14^13) = 3.4 x 2.14 = 7.276 = 10^-6 x 10^13 = 10^-6+13 = 10^7 = 7.276 x 10^7 |
| Dividing Scientific Notation: (3.4 x 10^-6) ÷ (2.14^13) | (3.4 x 10^-6) ÷ (2.14^13) = 3.4 ÷ 2.14 = 1.588785 = 10^-6 ÷ 10^13 = 10^-6-13 = 10^-19 = 1.588785 x 10^-19 |
| Multiplying and Dividing Scientific Notation: (2.3 x 10^-4) (6.4 x 10^12) / 4.2 x 10^10 | (2.3 x 10^-4) (6.4 x 10^12) / 4.2 x 10^10 = 2.3 x 6.4 = 14.72 = 10^4 x 10^12 = 10^-4+12 = 10^8 = 14.72 x 10^8 = 14.72 x 10^8 / 4.2 x 10^10 = 14.72 x 4.2 ≈ 3.5 = 10^8 ÷ 10^10 = 10^8-10 = 10^-2 = 3.5 x 10^-2 |
| Estimating Scientific Notation: (2.3 x 10^-4)(6.4 x 10^12) / 4.2 x 10^10 | (2.3 x 10^-4)(6.4 x 10^12) / 4.2 x 10^10 = (2 x 10^-4)(6 x 10^12) / 4 x 10^10 = 2 x 6 / 4 10^-4 x 10^12 / 10^10 = 12/4 10^-4+12/10^10 = 3 x 10^8/10^10 = 3 x 10^8-10 = 3 x 10^-2 |
| Estimating Scientific Notation: (13)(476)(52, 450)(975)(143) | (13)(476)(52, 450)(975)(143) = (1.3 x 10^1)(4.72 x 10^2)(5.245 x 10^4)(9.75 x 10^2)(1.43 x 10^2) = (1.3 x 4.72 x 5.245 x 9.75 x 1.43) (10^1 x 10^2 x 10^4 x 10^2 x 10^2) = (1.3 x 4.72 x 5.245 x 9.75 x 1.43) (10^1+2+4+2+2)... |
| Estimating Scientific Notation: (13)(476)(52, 450)(975)(143) - to be continued | = (1.3 x 4.72 x 5.245 x 9.75 x 1.43) (10^11) = (1.3 x 4.72 x 5.245 x 9.75 x 1.43) ≈ 452.5186 = (452.5186)(10^11) = 4.525186 x 10^2 x 10^11 = 4.525186 x 10^13 = 4.53 x 10^13 |
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