Adding, Multiplying, Dividing Fractions

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Questions and Answers

What is the decimal equivalent of 75%?

0.75 (correct)

75.0

7.5

0.075

Which of the following is the correct representation of 0.333... as a fraction?

1/4

3/11

1/3 (correct)

3/10

Round 5.678 to two significant figures.

6.0

5.7 (correct)

5.6

5.68

Which of the following represents the index notation for 'three times three times three times three'?

3^4 (C) Signup and view all the answers

Simplify the expression: m^8 ÷ m^2

m^6 (A) Signup and view all the answers

What is the value of 2^-3?

1/8 (A) Signup and view all the answers

Calculate the value of 8^(1/3).

2 (C) Signup and view all the answers

What is the HCF (Highest Common Factor) of 24 and 36?

12 (B) Signup and view all the answers

Express 60 as a product of primes.

2 x 2 x 3 x 5 (B) Signup and view all the answers

Write the number 0.0056 in standard form.

5.6 x 10^-3 (B) Signup and view all the answers

A shop sells apples for $2.50 per kilogram. How much would 3.5 kilograms of apples cost?

$8.75 (D) Signup and view all the answers

What is the best buy: 10 biscuits for $3.50 or 15 biscuits for $5.00?

15 biscuits for $5.00 (A) Signup and view all the answers

The exchange rate between US dollars and euros is 1 US dollar = 0.85 euros. How many euros would you get for 200 US dollars?

170 euros (D) Signup and view all the answers

A conversion graph shows that 5 miles is equivalent to 8 kilometers. Approximately how many kilometers is 12 miles?

19.2 kilometers (C) Signup and view all the answers

A conversion graph shows that 5 miles is equivalent to 8 kilometers. How many miles is 12 kilometers?

7.5 miles (C) Signup and view all the answers

Calculate the LCM (Lowest Common Multiple) of 12 and 18.

36 (A) Signup and view all the answers

Simplify the expression 6.75 x 10^4 x 2.4 x 10^2.

1.62 x 10^7 (B) Signup and view all the answers

What is the reciprocal of 3 1/2?

7/2 (D) Signup and view all the answers

What is the result of 1/3 ÷ 2/5?

5/6 (D) Signup and view all the answers

Which of the following is the correct way to find the lowest common multiple (LCM) of 6 and 8?

Find the prime factors of each number and multiply the highest power of each factor that appears. (B) Signup and view all the answers

What is the result of 2.5 x 0.4 ?

1.25 (C) Signup and view all the answers

What is 35% expressed as a fraction in its simplest form?

7/20 (B) Signup and view all the answers

What is the result of 5/8 + 1/4?

7/8 (B) Signup and view all the answers

Study Notes

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.

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Description

This quiz covers the essential operations of adding, multiplying, and dividing fractions. You will learn how to find common denominators, use reciprocals, and simplify fractions. Perfect for students looking to reinforce their fraction skills.