Adding, Multiplying, Dividing Fractions

Adding Fractions

• To add fractions, they must have the same denominator (a common denominator).
• Find the lowest common multiple (LCM) of the denominators.
• Create equivalent fractions with the LCM as the denominator.
• Add the numerators, keeping the denominator the same.
• Example: 2/5 + 1/4 = 8/20 + 5/20 = 13/20.
• To add mixed numbers, convert them to top-heavy fractions before finding the LCM and adding.

Multiplying Fractions

• Multiply the numerators together and the denominators together.
• Simplify the resulting fraction if possible.
• Example: 2/5 x 1/4 = (2x1)/(5x4) = 2/20 = 1/10

Dividing Fractions

• Multiply the first fraction by the reciprocal of the second fraction.
• The reciprocal of a fraction is found by flipping it over.
• Example: 7/15 ÷ 3/4 = 7/15 x 4/3 = 28/45

Reciprocals

• The reciprocal of a number is 1 divided by that number.
• Multiplying a number by its reciprocal always results in 1.
• Example: The reciprocal of 5 is 1/5, and 5 x 1/5 = 1.
• To find the reciprocal of a mixed number, convert it to a top-heavy fraction and then flip it over.

Multiplying Decimals

• Ignore the decimal points and multiply as whole numbers.
• Count the total number of digits after the decimal point in the original numbers.
• Place the decimal point in the answer so that there are the same number of digits after the decimal point as in the original numbers.
• Example: 2.854 x 8 = 22.832 (4 digits after the decimal point).

Dividing Decimals

• Multiply both the number being divided and the number dividing by the same power of 10 to make the divisor a whole number.
• Divide as usual.
• Example: 1.2 ÷ 0.04 = 120 ÷ 4 = 30.

Fractions, Decimals, and Percentages

• To change a fraction to a decimal, divide the numerator by the denominator.
• To change a decimal to a percentage, multiply by 100.
• To change a percentage to a fraction, write it over 100 and simplify.
• Example: 17/20 = 0.85 = 85%.
• Example: 0.315 = 31.5% = 315/1000 = 63/200.
• Example: 19% = 19/100 = 97/50.

Converting Fractions, Decimals, and Percentages

• Fractions can be represented as decimals by dividing the numerator by the denominator.
• Some fractions result in terminating decimals, which end after a certain number of digits.
• Other fractions result in recurring decimals, which have repeating patterns of digits that continue indefinitely.
• A percentage is a fraction out of 100. To convert a decimal to a percentage, multiply by 100.
• To convert a percentage to a decimal, divide by 100.

Converting Recurring Decimals to Fractions

• To convert a recurring decimal to a fraction, use algebra.
• Assign a variable, such as x, to represent the recurring decimal.
• Multiply the equation by a power of 10 so that the recurring portion of the decimal aligns.
• Subtract the original equation from the new equation to eliminate the recurring portion.
• Solve the resulting equation for x to get the fraction.

Rounding to Two Significant Figures

• Significant figures are the digits in a number that contribute to its precision.
• Identify the first two significant figures.
• Rounding up occurs if the third significant figure is 5 or greater.
• Rounding down occurs if the third significant figure is less than 5.
• When a decimal is being rounded, ignore leading zeroes.

Calculator Skills

• Familiarize yourself with the functions and buttons on your calculator.
• Use the fraction button to input fractions.
• Use the square root, cube root, and power buttons for calculations.
• Use the trigonometric functions (sine, cosine, tangent) for calculations.
• Understand how to convert between fractions and decimals on your calculator.

Estimation

• Round all numbers to one significant figure.
• Use the approximately equal to symbol (≈) to indicate that the answer is an estimate.
• Perform the calculations with the rounded numbers.

Best Buys

• Calculate the price per unit (e.g., price per biscuit) by dividing the total cost by the quantity.
• Compare the price per unit for different sizes to determine the best value.
• Alternatively, calculate the cost of buying an equal amount of units from each option to compare prices.

Exchange Rates

• To convert from one currency to another, multiply the amount in the original currency by the exchange rate.
• To convert from a foreign currency back to the original currency, divide the amount in the foreign currency by the exchange rate.

Conversion Graphs

• Conversion graphs can be used to compare distances, like miles and kilometers.
• They can also be used to convert money between currencies.
• To convert four miles into kilometers, find four on the horizontal axis, go vertically up to the line, then horizontally across to the vertical axis. In this case, four miles is approximately 6.4 kilometers.
• To convert eight kilometers into miles, find eight on the vertical axis, go horizontally across to the line, then vertically down to the horizontal axis. This shows that eight kilometers is equal to five miles.

Index Notation

• Index notation uses a number and a power to express repeated multiplication.
• Five times five times five is written as 5 cubed (5³).
• Two times two times two times two times two times two is written as 2 to the power of six (2⁶).

Laws of Indices

• Multiplying with the same base: When multiplying numbers with the same base, add the powers. (m³ x m⁴ = m⁷)
• Dividing with the same base: When dividing numbers with the same base, subtract the powers. (m⁸ ÷ m² = m⁶)
• Power of a power: When raising a power to another power, multiply the powers. (m³² = m⁶)

Negative Indices

• A negative index indicates the reciprocal of the base raised to the positive power.
• x^-n = 1/xⁿ
• Example: 2^-3 = 1/2³ = 1/8

Fractional Indices

• A fractional index represents a root.
• x^1/n = nth root of x
• Example: x^1/2 = square root of x
• x^m/n = nth root of x raised to the mth power
• To calculate a number with a fractional index, first take the root indicated by the denominator, then raise the result to the power indicated by the numerator.

LCM and HCF

• LCM (Lowest Common Multiple): The smallest number that is a multiple of both numbers.
• To find the LCM, list out the multiples of each number until you find the first number that is in both lists.
• HCF (Highest Common Factor): The largest number that divides evenly into both numbers.
• To find the HCF, list out all the factors of each number and identify the largest factor that is common to both lists.

Product of Primes

• Every whole number greater than 1 is either prime or a product of primes.
• Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
• To write a number as a product of primes, break it down into prime factors using a prime factor tree.
• Example: 60 = 2 x 3 x 2 x 5 = 2² x 3 x 5
• Index form makes it easy to find square numbers and cube numbers.
• To find the square number, share prime factors as evenly as possible, then add the extra ones to make the powers even.
• To find the cube number, share prime factors as evenly as possible, then add the extra ones to make the powers multiples of three.
• Product of Primes helps to find the LCM (Lowest Common Multiple) and HCF (Highest Common Factor)
• To find LCM and HCF, use a Venn Diagram with prime factors. Put shared factors in the middle and the rest in the respective circles.
• Multiply all prime factors in the Venn Diagram to obtain the LCM, and multiply the factors in the middle to find the HCF.

Standard Form

• Standard form is a way to write large and small numbers using powers of 10.
• Format: A number between 1 and 10, multiplied by 10 to a certain power (e.g., 3.4 x 10⁵)
• To write a whole number in standard form, choose the number between 1 and 10 and count the number of zeros.
• For example, 7000 = 7 x 10³ (three zeros = 10³).
• For decimals, count the number of decimal places moved to the right to get the power of 10 (negative for smaller numbers).
• Example: 0.018 = 1.8 x 10^-2 (decimal point moved two places to the right).
• To write a number almost in standard form (e.g., 562.8 x 10⁵)
• 1. Divide: Divide the number outside the power of 10 (562.8) by 100 to get 5.628.
• 2. Compensation: Compensate this divide by multiplying the 10⁵ by 100 to find 5.628 x 10⁷.
• The final number should be between 1 and 10 multiplied by 10 to a power.