Math Class: Rounding and Factors

Rounding to One Significant Figure

• Rounding 6.32 to one significant figure results in 6.
• Rounding 6.51 to one significant figure results in 7.
• Rounding 0.503 to one significant figure results in 0.5.
• Dividing by 0.5 doubles the value.

Prime Factorization

• The prime factorization of 180 is 2² x 3² x 5.

Lowest Common Multiple (LCM)

• The LCM of 15 and 40 is 120.

Highest Common Factor (HCF)

• The HCF of 72 and 90 is 18.
• The HCF can be determined using prime factorization: find the common prime factors and multiply them together.

Inequalities on a Number Line

• To represent the inequality -2 ≤ x < 4 on a number line, draw a closed circle at -2 and an open circle at 4, connecting the two with a line.
• The closed circle at -2 indicates that -2 is included in the solution.
• The open circle at 4 indicates that 4 is not included in the solution.
• To represent the inequality x ≥ 1 on a number line, draw a closed circle at 1 and extend a line to the right, indicating that all values greater than or equal to 1 are included.

Percentages

• To calculate a percentage of an amount, first find 10% and then use that value to calculate other percentages.
• For example, 20% of 640 can be found by first finding 10% (64) and then multiplying it by 2 (128).

Reverse Percentages

• To find the original price after a sale, recognize the relationship between the sale price and the percentage it represents.
• For example, if a phone has been reduced by 20%, the sale price represents 80% of the original price.
• To find the original price, divide the sale price by the percentage it represents (80%) to find 1%, then multiply by 100 to find 100%.

Increases and Decreases in Value

• To find the original value after an increase, recognize the relationship between the increase and the percentage it represents.
• For example, if a house increases in value by 3%, the increase represents 3% of the original value.
• To find the original value, divide the increase by the percentage it represents (3%) to find 1%, then multiply by 100 to find 100%.

Simple Interest

• Simple interest is calculated on the original amount invested and does not accumulate over time.
• To find the amount of simple interest, multiply the principal amount by the interest rate and the number of years.
• The total amount in the account after interest is added is the principal plus the interest earned.

Compound Interest

• Compound interest is calculated on the principal amount plus the accumulated interest from previous periods.
• To calculate compound interest, multiply the principal amount by the interest rate (plus 1) to the power of the number of years.
• Compound interest is the difference between the final amount and the original principal.

Depreciation

• Depreciation refers to the decrease in an asset's value over time.
• To calculate depreciation, multiply the original value by the depreciation rate (plus 1) to the power of the number of years.
• The depreciated value is the original value minus the total depreciation.

Multiplying Fractions

• To multiply fractions, multiply the numerators and multiply the denominators.
• Simplify the resulting fraction to its simplest form.

Dividing Fractions

• To divide fractions, keep the first fraction the same, flip the second fraction, and multiply.
• Simplify the resulting fraction to its simplest form.

Adding and Subtracting Fractions

• To add or subtract fractions with different denominators, find a common denominator by multiplying the numerator and denominator of each fraction by a suitable factor.
• Add or subtract the numerators, keeping the common denominator.
• Simplify the resulting fraction.

Working with Fractions and Whole Numbers

• When adding, subtracting, multiplying, or dividing fractions with whole numbers, convert the whole number to a fraction with a denominator of 1.

Standard Form (Scientific Notation)

• Standard form represents a number in the form a x 10^n, where a is between 1 and 10, and n is an integer.
• To express a number in standard form, move the decimal point to create a number between 1 and 10, and count the number of places the decimal was moved.
• If the original number is larger than 1, the power of 10 will be positive.
• If the original number is smaller than 1, the power of 10 will be negative.

Writing Numbers in Standard form

• A number multiplied by 10 to the power of 5 becomes a bigger number.
• Move the decimal point 5 places to the right and fill in zeros.
• 2.3 x 10⁵ = 230,000
• A number multiplied by 10 to the power of -3 becomes a smaller number.
• Move the decimal point 3 places to the left and fill in zeros.
• 18.04 x 10^-3 = 0.01804

Index Laws

• When multiplying powers with the same base, add the exponents.
• 3² x 3³ = 3⁵
• When dividing powers with the same base, subtract the exponents.
• 3⁷ ÷ 3³ = 3⁴
• A negative exponent indicates the reciprocal of the base.
• (5)^-1 = 1/5
• A fractional exponent represents a root.
• 64^1/2 = √64 = 8

Combinations

• To find the number of combinations, multiply the number of choices for each option.
• For eight different bikes and ten paint colors, there are 8 x 10 = 80 combinations.

Error Interval

• Error interval is the range of possible values for a rounded number.
• When rounding to one decimal place, add or subtract 0.05.
• 6.4 rounded to one decimal place has an error interval of 6.35 ≤ x < 6.45
• Truncation means the number is chopped off at a specific decimal place.
• 6.4 truncated to one decimal place has an error interval of 6.4 ≤ x < 6.5

Using a Calculator

• Use the calculator to find square roots (√), cube roots (∛), squares (^2), cubes (^3), fractions, sine (sin), cosine (cos), and tangent (tan).
• To find the cube root (∛) on most calculators, press "shift" followed by the "√" button.
• To enter a fraction, use the "fraction" button on the calculator.
• When calculating trigonometric functions, do not enter a degree symbol, the calculator understands degrees.
• Write down all the digits on the calculator display before rounding.

Expanding Brackets

• Multiply everything inside the bracket by the term outside the bracket.
• 3b (3b + 7) = 3b x 3b + 3b x 7 = 9b² + 21b
• Remember to pay attention to signs - a negative sign outside the bracket affects the sign of the terms inside.
• (3y – 2)(5y – 4y²) = 15y – 12y² - 10y² + 8y³ = 8y³ – 22y² + 15y

Factorizing

• Take out a common factor from all terms and put them inside brackets.
• 3x + 15 = 3(x + 5)
• To factorize fully, look for the highest common factor of coefficients and variables.
• 35x – 21x² = 7x(5 – 3x)

Factorizing Quadratics

• Find two numbers that multiply to give the constant term and add to give the coefficient of the x term.
• x² + 7x + 12 = (x + 3)(x + 4)
• If the coefficient of the x term is negative, the two numbers must sum to a negative value.
• x² - 3x - 28 = (x + 4)(x - 7)

Solving Equations

• Solve for the value of the unknown variable by performing inverse operations on both sides of the equation.
• 3x + 5 = 26
  • 3x = 21
  • x = 7
• If the variable appears on both sides of the equation, move it to one side by performing opposite operations.
• 5x - 2 = 21
  • 5x = 23
  • x = 23/5

Rearranging Formulas

• To make 'x' the subject, we isolate 'x' on one side of the equation.
• To remove a constant term, subtract it from both sides.
• To reverse a division, multiply both sides by the divisor.
• We can rewrite the formula with 'x' on the left or right side without changing its meaning.

Solving Equations with Fractions

• To eliminate a fraction, multiply both sides by the denominator.
• Expanding multiplication can simplify the expression.

Substituting Values

• When substituting values for variables, always enclose them in brackets to avoid errors in calculations.

Solving Simultaneous Equations

• To solve simultaneous equations, we aim to eliminate one variable.
• Make the coefficients of one variable the same in both equations by multiplying each equation by an appropriate factor.
• If the signs in front of the variable to be eliminated are different, add the equations together.
• If the signs are the same, subtract the equations.
• Solve the remaining equation for the remaining variable.
• Substitute the obtained value back into one of the original equations to solve for the other variable.

Word Problems with Equations

• Convert word problems into equations by representing unknown quantities with variables.
• Set up the equations based on the relationships described in the problem.
• Solve the equations using the same techniques applied for simultaneous equations.

Finding Perimeter

• Determine the perimeter of each shape using the given lengths.
• Translate the relationship between perimeters into an equation.
• Solve the equation to find the value of the unknown variable.
• Substitute the value back into the perimeter expressions to find the actual perimeters.

Angles in a Pentagon

• The sum of interior angles in a pentagon is 540 degrees.
• Create an equation by adding the expressions for each angle and setting the sum equal to 540 degrees.
• Solve the equation for the unknown variable.
• Substitute the value back into the expression for the angle that needs to be determined.

Solving Inequalities

• Treat inequalities similar to equations, performing the same operations on both sides.
• The inequality symbol remains unchanged throughout the solving process.

Finding the nth Term of a Sequence

• Determine the common difference between consecutive terms.
• Write the nth term as a multiple of the common difference multiplied by 'n', plus or minus a constant.
• The constant can be found by observing the difference between the nth term and the corresponding term in the times table multiplied by the common difference.
• Substitute the desired value of 'n' into the nth term formula to find the specific term.

Explaining Sequence Membership

• Check if all terms in the sequence follow the same pattern or rule.
• Use the nth term formula to determine if the given term would be generated by the formula.
• If the formula produces a non-whole number result, the given term is not part of the sequence.

Graphing Linear Functions

• Create a table of x and y values using the linear equation.
• Choose a range of 'x' values, usually including negative and positive values and zero.
• Substitute each 'x' value into the equation and calculate the corresponding 'y' value.
• Plot the points (x, y) on a graph.
• Draw a straight line through the plotted points to represent the graph of the linear function.

Understanding Graph Equations

• The equation of a straight line is y = mx + c
• The value of 'm' represents the gradient of the line
• The value of 'c' represents the y-intercept, where the line crosses the y-axis

Creating a Linear Graph from its Equation

• Use a table to find the coordinates of points that satisfy a linear equation
• Substitute different values for 'x' into the equation to find corresponding values for 'y'
• Plot the coordinates on a graph and join them with a straight line

Finding the Equation of a Line from a Graph

• Identify any two whole number coordinates on the line
• Calculate the gradient: divide the change in y-values by the change in x-values
• Find the y-intercept, where the line crosses the y-axis
• Substitute the calculated gradient (m) and y-intercept (c) into the equation y = mx + c

Parallel Lines

• Parallel lines have the same gradient (m) in their equations (y = mx + c)

Working with Ratios

• A ratio represents a relationship between two or more quantities
• To find the value of one part of the ratio, divide the total value by the sum of the ratio parts
• Multiply the value of one part by the relevant ratio value to find the individual values

Currency Conversions

• An exchange rate tells you how much one currency is worth in another currency
• To convert from one currency to another, multiply or divide by the exchange rate
• Divide to convert from higher value currency to lower value currency
• Multiply to convert from lower value currency to higher value currency

Working with Recipes

• A recipe can be scaled up or down by multiplying or dividing the ingredient quantities by a factor
• To make a larger or smaller quantity of a recipe, adjust the ingredient amounts proportionally
• To find the amount of a specific ingredient needed for a certain number of people, divide the total amount of the ingredient by the original number of servings and multiply by the desired number of servings

Calculating Best Value for Money

• Compare the cost of different products by calculating the price per unit (e.g., price per kilogram, price per liter)
• Divide the total cost by the total quantity to find the unit price
• The product with the lowest unit price offers better value for money

Density

• Density is a measure of mass per unit volume
• To find the density of a substance, divide the mass by the volume: Density = Mass / Volume
• Density is measured in units such as grams per cubic centimeter (g/cm³)

Working with Mixtures

• When mixing substances with different densities, the density of the mixture is not simply an average
• To calculate the density of a mixture, first, you must find the combined mass and volume:
  • Total mass: Sum of the individual masses of each component
  • Total volume: Sum of the individual volumes of each component
• Then, divide the combined mass by the combined volume to find the density of the mixture.

This quiz covers key concepts in mathematics such as rounding to one significant figure, prime factorization, and determining the lowest common multiple and highest common factor. Additionally, it includes representing inequalities on a number line. Ideal for reinforcing your understanding of these fundamental topics.