jhtgrefwds

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson
Download our mobile app to listen on the go
Get App

Questions and Answers

Which number system is closed under both addition and multiplication?

  • Rational Numbers
  • Whole Numbers
  • Integers
  • Natural Numbers (correct)

What is the identity element for addition in the system of whole numbers?

  • -1
  • 1
  • 10
  • 0 (correct)

Which of the following is an example of an irrational number?

  • $\sqrt{4}$
  • -3
  • $\pi$ (correct)
  • $ rac{1}{2}$

Estimating and rounding off numbers are techniques primarily used to:

<p>Simplify calculations (A)</p> Signup and view all the answers

What is the purpose of 'compensating' when estimating calculations?

<p>To correct errors introduced by rounding (A)</p> Signup and view all the answers

What is the definition of a common multiple of two or more numbers?

<p>A number that is a multiple of each of them. (D)</p> Signup and view all the answers

What is the purpose of prime factorization?

<p>To express a number as a product of its prime factors (C)</p> Signup and view all the answers

Which method is used to determine both the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two numbers?

<p>Prime factorization (C)</p> Signup and view all the answers

If Moeneba collects apples at a rate of 5 apples per minute, how many apples does she collect in 15 minutes?

<p>75 (A)</p> Signup and view all the answers

In a biscuit recipe, 5 parts of flour are mixed with 2 parts of oatmeal and 1 part of cocoa powder. What is the ratio of flour to oatmeal?

<p>5:2 (A)</p> Signup and view all the answers

What formula is used to calculate average speed?

<p>Average speed = Distance / Time (C)</p> Signup and view all the answers

What is the difference between simple interest and compound interest?

<p>Simple interest is calculated only on the principal, while compound interest is calculated on the principal plus accumulated interest. (C)</p> Signup and view all the answers

Adding a negative number to a positive number is equivalent to:

<p>Subtracting the numbers (C)</p> Signup and view all the answers

What is the result of subtracting a negative number from a positive number?

<p>The two numbers are added (A)</p> Signup and view all the answers

According to the distributive property, what is the equivalent expression for $a(b + c)$?

<p>$ab + ac$ (C)</p> Signup and view all the answers

What is the result of multiplying two negative integers?

<p>A positive integer (C)</p> Signup and view all the answers

What happens when a number is added to its additive inverse?

<p>The result is always 0 (D)</p> Signup and view all the answers

Which of the following statements is always true when (x) is a real number?

<p>$x^2 \ge 0$ (D)</p> Signup and view all the answers

How does subtracting a negative number affect the original number?

<p>It increases the original number (C)</p> Signup and view all the answers

What is the primary reason for the invention of negative numbers?

<p>To provide solutions for equations that involve subtracting a larger number from a smaller number (C)</p> Signup and view all the answers

If $a$ is a positive number and $b$ is a negative number, which of the following is always negative?

<p>$a imes b$ (A)</p> Signup and view all the answers

What is the value of $(-1)^{-1}$?

<p>-1 (A)</p> Signup and view all the answers

What does $a^{-n}$ represent?

<p>$ rac{1}{a^n}$ (D)</p> Signup and view all the answers

Based on the laws of exponents, what is the simplified form of $ rac{a^5}{a^2}$?

<p>$a^3$ (B)</p> Signup and view all the answers

A number in scientific notation is written as $3.2 imes 10^{-4}$. What is this number in decimal form?

<p>0.00032 (C)</p> Signup and view all the answers

Simplify the expression: $(2^3 imes 2^2) \div 2^4$.

<p>2 (D)</p> Signup and view all the answers

If $a^m = a^n$, what must be true?

<p>$m = n$ (B)</p> Signup and view all the answers

What is $5^0$?

<p>1 (D)</p> Signup and view all the answers

Evaluate $(-3)^3$.

<p>-27 (B)</p> Signup and view all the answers

Given that $x$ is an integer and $x < 0$, which of the following expressions will always yield a positive result?

<p>$x^2$ (A)</p> Signup and view all the answers

Solve for $x$: $2^{x+1} = 8$.

<p>2 (B)</p> Signup and view all the answers

What is the value of $x$ if $5^x = rac{1}{25}$?

<p>-2 (C)</p> Signup and view all the answers

If the cost price of an item is $C$, and it is sold for $S$, where $S < C$, then there is a:

<p>Loss (D)</p> Signup and view all the answers

A journey is completed in two sections. The first section covers 120 km at an average speed of 60 km/h, and the second section covers 180 km at an average speed of 90 km/h. What is the total time for the journey?

<p>5 hours (D)</p> Signup and view all the answers

Consider two numbers, $x$ and $y$. If $x$ is a positive integer and $y$ is a negative integer, which of the following expressions results in the largest value?

<p>$x - y$ (D)</p> Signup and view all the answers

A store offers a 20% discount on all items. If a customer buys two items, one priced at $50 and another at $30, what is the total amount the customer will pay after the discount?

<p>$64 (D)</p> Signup and view all the answers

Determine which of the following numbers, when raised to any positive integer power, will always result in a number less than or equal to the original number?

<p>0.99 (D)</p> Signup and view all the answers

Which number system includes both rational and irrational numbers?

<p>Real numbers (D)</p> Signup and view all the answers

What does 'prime factorization' involve?

<p>Expressing a number as a product of its prime factors. (A)</p> Signup and view all the answers

If $a$ and $b$ are integers, which expression always results in an integer?

<p>$a + b$ (B)</p> Signup and view all the answers

What is the result of adding an integer to its additive inverse?

<p>0 (D)</p> Signup and view all the answers

According to the laws of exponents, what is the simplified form of $(a^3)^4$?

<p>$a^{12}$ (C)</p> Signup and view all the answers

What is the value of $(-10)^0$?

<p>1 (B)</p> Signup and view all the answers

What is the value of $x$ in the equation $3^{x} = 81$?

<p>4 (A)</p> Signup and view all the answers

What is the main difference between simple interest and compound interest?

<p>Simple interest is calculated on the principal, while compound interest is calculated on the principal plus accumulated interest. (D)</p> Signup and view all the answers

What is the result of the expression $5 - (-3)$?

<p>8 (B)</p> Signup and view all the answers

Which expression is equivalent to $x(y - z)$ according to the distributive property?

<p>$xy - xz$ (A)</p> Signup and view all the answers

What is the sign of the result when a positive integer is divided by a negative integer?

<p>Negative (A)</p> Signup and view all the answers

What is the value of $a^{-3}$?

<p>$\frac{1}{a^3}$ (B)</p> Signup and view all the answers

What is the decimal form of the number $6.2 imes 10^{5}$?

<p>620,000 (B)</p> Signup and view all the answers

If $b^x = b^y$ and $b \neq 0, 1, -1$, what can you conclude about $x$ and $y$?

<p>$x = y$ (C)</p> Signup and view all the answers

The price of an item is marked up by 25%, but then it is sold at a 10% discount off the marked price. What is the overall percentage profit?

<p>12.5% (A)</p> Signup and view all the answers

In scientific notation, which of the following represents 0.0000075?

<p>$7.5 \times 10^{-6}$ (C)</p> Signup and view all the answers

If a store increases the price of an item by 15% and then decreases it by 10%, what is the net percentage change in the price?

<p>Increase of 3.5% (C)</p> Signup and view all the answers

If $x < 0$, which of the following expressions is always positive?

<p>$x^2$ (B)</p> Signup and view all the answers

If the number 0.000047 is expressed in scientific notation as $4.7 \times 10^n$, what is the value of $n$?

<p>-5 (D)</p> Signup and view all the answers

Which of the following numbers is NOT a whole number?

<p>1/2 (A)</p> Signup and view all the answers

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

<p>12 (C)</p> Signup and view all the answers

If the exchange rate is $1 = 0.85, how many euros would you get for $200, after a commission of 2% is deducted?

<p>166.60 (A)</p> Signup and view all the answers

If a number $x$ is decreased by 20% and then increased by 25%, what is the net percentage change from the original value?

<p>No change (B)</p> Signup and view all the answers

A car travels 240 km in 3 hours. If the car maintains the same average speed, how long will it take to travel an additional 320 km?

<p>4 hours (C)</p> Signup and view all the answers

A retailer buys an item for $80 and marks it up by 40%. If they then offer a discount of 15% on the marked price, what is the final selling price?

<p>$92.00 (B)</p> Signup and view all the answers

Two cyclists start at the same point and travel in opposite directions. One cyclist travels at 20 km/h and the other at 25 km/h. How far apart are they after 2.5 hours?

<p>112.5 km (C)</p> Signup and view all the answers

What is the value of $x$ in the equation $\frac{2^x}{2^3} = 16$?

<p>7 (A)</p> Signup and view all the answers

A jacket is priced at $120. If a customer has a coupon for 15% off and pays with a credit card that offers an additional 5% discount, what is the final price they pay?

<p>$96.90 (B)</p> Signup and view all the answers

John invests $5,000 in a simple interest account with an annual interest rate of 4%. How much interest will he earn after 3 years?

<p>$600 (A)</p> Signup and view all the answers

Consider the expression $x^2 + 2x + 1$. What is its value when $x = -1$?

<p>0 (B)</p> Signup and view all the answers

If $f(x)=x^2-4x+3$, find $f(2)$.

<p>-1 (C)</p> Signup and view all the answers

If $f(x) = 3x^3 - 2x + 5$ and $g(x) = x^2 - x + 2$, what is the value of $(f+g)(1)$?

<p>9 (C)</p> Signup and view all the answers

Which set of numbers includes 0 and all counting numbers?

<p>Whole numbers (C)</p> Signup and view all the answers

Which of the following number systems is NOT always closed under subtraction?

<p>Natural numbers (B)</p> Signup and view all the answers

Which number, when added to any whole number, leaves the whole number unchanged?

<p>0 (D)</p> Signup and view all the answers

Which of the following correctly describes a key attribute of integers?

<p>Every integer has an additive inverse. (A)</p> Signup and view all the answers

Which of the following numbers cannot be expressed as a fraction of two integers?

<p>$\pi$ (B)</p> Signup and view all the answers

What set encompasses both rational and irrational numbers?

<p>Real Numbers (B)</p> Signup and view all the answers

What is the primary goal when estimating calculations?

<p>To get an approximate value quickly (C)</p> Signup and view all the answers

Why is 'compensating' important in estimation?

<p>It corrects errors introduced by rounding. (C)</p> Signup and view all the answers

What are consecutive multiples of a number?

<p>Multiples that follow each other in a sequence. (B)</p> Signup and view all the answers

What is the Lowest Common Multiple (LCM) of two or more numbers?

<p>The smallest number that is a multiple of each of the numbers. (C)</p> Signup and view all the answers

How is the Highest Common Factor (HCF) determined using prime factorization?

<p>By identifying and multiplying the common prime factors of the numbers. (D)</p> Signup and view all the answers

If a mixture requires a ratio of 3 parts sand to 1 part cement, and you have 6 parts of sand, how much cement do you need?

<p>2 parts (C)</p> Signup and view all the answers

A car travels 200 km in 4 hours. What formula would you use to find its average speed?

<p>Distance / Time (C)</p> Signup and view all the answers

What financial concept involves paying a deposit followed by monthly installments?

<p>Hire Purchase (D)</p> Signup and view all the answers

Adding a negative number to a positive number is equivalent to which operation?

<p>Subtraction (A)</p> Signup and view all the answers

What is the result of multiplying a positive integer by a negative integer?

<p>Always negative (B)</p> Signup and view all the answers

If a number is added to its additive inverse, what is the result?

<p>Zero (B)</p> Signup and view all the answers

What is the result of dividing a negative number by another negative number?

<p>A positive number (B)</p> Signup and view all the answers

Express the number 56,700 in scientific notation.

<p>$5.67 \times 10^4$ (D)</p> Signup and view all the answers

Solve for $x$: $3^{x} = 27$.

<p>3 (A)</p> Signup and view all the answers

Simplify: $\frac{5^6 \times 5^{-2}}{5^2}$

<p>$5^2$ (B)</p> Signup and view all the answers

If $p$ and $q$ are integers such that $p < 0$ and $q > 0$, which expression is always negative?

<p>$p \times q$ (D)</p> Signup and view all the answers

What is the result of subtracting a negative number from another negative number?

<p>Could be positive or negative (B)</p> Signup and view all the answers

If $x$ is a real number, which of the following is always non-negative?

<p>$x^2$ (A)</p> Signup and view all the answers

Which expression is equivalent to $-(a - b)$?

<p>$-a + b$ (A)</p> Signup and view all the answers

Given the expression $(-1)^n$, under what condition will the result always be -1?

<p>When n is an odd integer (D)</p> Signup and view all the answers

If $f(x) = x^2 - 3x + 2$, for what values of $x$ does $f(x) = 0$?

<p>x = 1, x = 2 (B)</p> Signup and view all the answers

Let $a, b, c$ be integers such that $a + b + c = 0$. What is the minimum possible value of $a^2 + b^2 + c^2$ if at least one of $a, b, c$ is nonzero?

<p>2 (B)</p> Signup and view all the answers

Flashcards

Natural Numbers

Numbers used for counting, starting from 1: 1, 2, 3,...

Whole Numbers

Natural numbers plus zero: 0, 1, 2, 3,...

Integers

Extends whole numbers to include negative numbers.

Rational Numbers

Numbers that can be expressed as a fraction of two integers.

Signup and view all the flashcards

Irrational Numbers

Numbers that cannot be expressed as a fraction of two integers.

Signup and view all the flashcards

Real Numbers

Includes both rational and irrational numbers.

Signup and view all the flashcards

Estimating

Getting close to an answer without precise calculations.

Signup and view all the flashcards

Rounding Off

Rounding off numbers to simplify calculations.

Signup and view all the flashcards

Compensating for Errors

Correcting errors introduced by rounding.

Signup and view all the flashcards

Consecutive Multiples

Numbers obtained by multiplying a number by 1, 2, 3, and so on.

Signup and view all the flashcards

Common Multiple

A number that is a multiple of each of two or more given numbers.

Signup and view all the flashcards

Lowest Common Multiple (LCM)

The smallest multiple shared by two or more numbers.

Signup and view all the flashcards

Prime Factorization

Expressing a number as a product of its prime factors.

Signup and view all the flashcards

Highest Common Factor (HCF)

The largest number that divides each of two or more numbers without a remainder.

Signup and view all the flashcards

Ratio

Comparison of two quantities.

Signup and view all the flashcards

Rate

A measure, quantity, or frequency, typically one measured against some other quantity or measure.

Signup and view all the flashcards

Proportion

An equation stating that two ratios are equal.

Signup and view all the flashcards

Discount

Percentage reduction from the original price.

Signup and view all the flashcards

Profit

The amount gained from a sale.

Signup and view all the flashcards

Loss

The amount lost from a sale.

Signup and view all the flashcards

Cost Price

Original cost of an item.

Signup and view all the flashcards

Marked Price

The price at which an item is listed for sale.

Signup and view all the flashcards

Selling Price

The price at which an item is actually sold.

Signup and view all the flashcards

Hire Purchase

Paying a deposit followed by monthly payments.

Signup and view all the flashcards

Simple Interest

Interest calculated only on the principal amount.

Signup and view all the flashcards

Compound Interest

Interest calculated on the principal plus accumulated interest.

Signup and view all the flashcards

Currency Exchange

Converting one currency to another.

Signup and view all the flashcards

Commission

The additional fee charged for currency exchange.

Signup and view all the flashcards

Adding a Negative

Adding a negative number is the same as subtracting its positive counterpart: (a + (-b) = a - b)

Signup and view all the flashcards

Subtracting a Negative

Subtracting a negative number becomes addition: (a - (-b) = a + b)

Signup and view all the flashcards

Additive Inverse

Integer that, when added to (a), results in 0: (a + (-a) = 0)

Signup and view all the flashcards

Exponents

Shorthand for repeated multiplication: (5 \times 5 \times 5 = 5^3)

Signup and view all the flashcards

Exponent Law: Multiplication

Multiplying powers with the same base: (a^m \times a^n = a^{m+n})

Signup and view all the flashcards

Exponent Law: Division

Dividing powers with the same base: (a^m \div a^n = a^{m-n})

Signup and view all the flashcards

Exponent Law: Power of a Power

Raising a power to a power: ((a^m)^n = a^{m \times n})

Signup and view all the flashcards

Whole Number System

The system of numbers including natural numbers and zero.

Signup and view all the flashcards

Identity Element for Addition

When 0 is added to any number, the number remains unchanged.

Signup and view all the flashcards

Compensating

The process of adjusting calculations to account for initial approximations.

Signup and view all the flashcards

Average Speed Formula

Average speed is total distance covered divided by total time taken.

Signup and view all the flashcards

Hire Purchase Interest

The difference between the total hire purchase price and the cash price.

Signup and view all the flashcards

Pos x Neg = ?

The product of a positive and a negative number is always negative.

Signup and view all the flashcards

Neg x Neg = ?

The product of two negative numbers is positive.

Signup and view all the flashcards

Distributive Property

The property stating a(b + c) = ab + ac.

Signup and view all the flashcards

Pos ÷ Neg = ?

The quotient of a positive number and a negative number is negative.

Signup and view all the flashcards

Neg ÷ Neg = ?

The quotient of two negative numbers is positive.

Signup and view all the flashcards

Negative Numbers

Numbers less than zero.

Signup and view all the flashcards

Scientific Notation

Moving the decimal point to express very large or small numbers using powers of 10.

Signup and view all the flashcards

Scientific Notation (Large Numbers)

Writing very large numbers in the form ± a × 10^n, where n is positive.

Signup and view all the flashcards

Scientific Notation (Small Numbers)

Writing very small numbers in the form ± a × 10^n, where n is negative.

Signup and view all the flashcards

Decimal Form Conversion

Moving the decimal point according to the exponent of 10 to return to decimal form.

Signup and view all the flashcards

Counting Numbers

Numbers such as 1, 2, 3...

Signup and view all the flashcards

Simplifying Exponential Expressions

Removing numbers from both sides to equalise an expressions with exponential forms.

Signup and view all the flashcards

Solving Exponential Expressions

When exponent are the reciprocal of each other.

Signup and view all the flashcards

Word Problems Involving Integers

Used in real-world situation to understand the numbers, depths, finances, and temperature.

Signup and view all the flashcards

Closure Property

The system of natural numbers is closed under addition and multiplication because performing these operations on natural numbers always results in another natural number.

Signup and view all the flashcards

The Purpose of Integers

A number system that includes negative numbers, allowing solutions to subtraction problems where a larger number is subtracted from a smaller number.

Signup and view all the flashcards

Compensating for Rounding

Errors introduced by rounding off numbers during estimation can be corrected by adjusting the final result to account for the initial rounding.

Signup and view all the flashcards

Total HP Price

The deposit plus the total of all monthly installments paid.

Signup and view all the flashcards

Exponent Law: Product to a Power

When raising a product to a power, distribute the power to each factor:

(a \times b)^n = a^n \times b^n

Signup and view all the flashcards

Square of a Number

A number multiplied by itself. Denoted as x^2 .

Signup and view all the flashcards

Cube of a Number

A number multiplied by itself twice. Denoted as x^3 .

Signup and view all the flashcards

Square Root

The positive square root of a number x, denoted (\sqrt{x}).

Signup and view all the flashcards

Anything to the power 0.

If a number to the power of 0.

Signup and view all the flashcards

Study Notes

Properties of Numbers

  • Natural numbers are used for counting and begin with 1.
  • Adding or multiplying natural numbers results in another natural number.
  • Subtracting or dividing natural numbers does not always result in a natural number.
  • Whole numbers include natural numbers and 0.
  • Adding 0 to a number does not change the number (Identity Element for Addition).
  • Integers extend whole numbers to include negative numbers.
  • The sum of two integers can be zero, and each integer has an additive inverse.
  • Rational numbers can be expressed as a quotient of two integers.
  • Irrational numbers cannot be expressed as a fraction of two integers (e.g., (\sqrt{2}), (\sqrt{5}), (\pi)).
  • Real numbers include both rational and irrational numbers.

Calculations with Whole Numbers

  • Estimating involves approximating answers without precise calculations.
  • Rounding off simplifies calculations.
  • Compensation corrects errors introduced by rounding.
  • Adding, multiplying, and subtracting can be performed using column methods.
  • Long division involves dividing numbers step-by-step.

Multiples and Factors

  • Consecutive multiples are obtained by multiplying a number by 1, 2, 3, etc.
  • A common multiple is a number that is a multiple of two or more numbers.
  • The Lowest Common Multiple (LCM) is the smallest common multiple of two or more numbers.
  • Prime factorization expresses a number as a product of its prime factors.
  • The Highest Common Factor (HCF) is the largest number that divides each number without a remainder.
  • To find the LCM, multiply all prime factors of both numbers, without repeating.
  • To find the HCF, multiply the common prime factors of the numbers.

Solving Problems about Ratio, Rate, and Proportion

  • Rate example: Moeneba collects about 5 apples per minute.
  • Ratios compare collection rates between different people (e.g., Garth collects about 12, and Kate collects about 15 apples per minute).
  • Biscuit recipe example: 5 parts flour, 2 parts oatmeal, 1 part cocoa powder.
  • Average speed = Distance / Time.
  • Distance = Average speed × Time.
  • Time = Distance / Average speed.

Solving Problems in Financial Contexts

  • Percentage calculations determine discounts, profits, and losses.
  • Key terms: cost price, marked price, selling price.
  • Hire purchase involves a deposit and monthly instalments.
    • Total HP price is the deposit plus the total instalments.
    • Interest is the difference between the HP price and the cash price.
  • Simple interest is calculated on the principal amount.
  • Compound interest adds interest to the principal each year.
  • Currency exchange converts one currency to another based on the exchange rate, with potential commissions.

Adding and Subtracting with Integers

  • Adding a negative number is equivalent to subtracting the corresponding positive number.
  • Subtracting a negative number is equivalent to adding the corresponding positive number.
  • Subtracting a larger number from a smaller number results in a negative number.
  • Terms:
    • ( a + (-b) = a - b )
    • ( a - (-b) = a + b )
    • ( a + (-a) = 0 )
    • ( a - (-a) = a + a )

Multiplying and Dividing with Integers

  • The product of two positive numbers is positive.
  • The product of a positive number and a negative number is negative.
  • The product of two negative numbers is positive.
  • Distributive Property: ( a(b + c) = ab + ac ).
  • The quotient of a positive number and a negative number is negative.
  • The quotient of two negative numbers is positive.
  • Adding an integer has the same effect as subtracting its additive inverse.
  • Subtracting an integer has the same effect as adding its additive inverse.

Powers, Roots, and Word Problems

  • The square of ( x ) is ( x^2 ).
  • The cube of ( x ) is ( x^3 ).
  • The square root of ( x ) is ( \sqrt{x} ).
  • Positive and negative numbers can have square and cube roots.
  • Integers are used in real-world problems such as temperature changes and financial transactions.

Integers

  • Negative numbers (-7, -500, -3/4, -3.46) are additive inverses of whole numbers.
  • When a larger number is subtracted from a smaller one, the result is negative (e.g., 5 − 12 = −7).
  • Properties of Negative Numbers:
    • Adding a Negative Number: Equivalent to subtracting the corresponding positive number.
    • Subtracting a Negative Number: Equivalent to adding the corresponding positive number.
    • Product of a Positive and a Negative Number: The result is a negative number.

Exponents

  • Exponents indicate repeated multiplication (e.g., (5 \times 5 \times 5 = 5^3)).
  • In mixed operations, powers are calculated before multiplication and division.
  • Laws of Exponents:
    • (a^m \times a^n = a^{m+n})
    • (a^m \div a^n = a^{m-n})
    • ((a^m)^n = a^{m \times n})
    • ((a \times b)^n = a^n \times b^n)
    • (a^0 = 1)

Order of Operations

  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
  • Negative exponents: (a^{-n} = \frac{1}{a^n}).

Scientific Notation

  • Expresses numbers as ( \pm a \times 10^n), where (1 \leq a < 10) and (n) is an integer.
  • Large numbers: Move the decimal point to the left; (n) is positive.
  • Small numbers: Move the decimal point to the right; (n) is negative.
  • Simplifying Exponential Expressions: Use the laws of exponents.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team
Use Quizgecko on...
Browser
Browser