Rational Numbers: Arithmetic and Problem Solving

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Questions and Answers

Which of the following statements is always true regarding an integer and its opposite?

  • The quotient of an integer and its opposite is always 1.
  • The product of an integer and its opposite is always positive.
  • The integer and its opposite are equal.
  • The sum of an integer and its opposite is always zero. (correct)

All rational numbers are integers.

False (B)

What is the result of dividing any non-zero integer by its multiplicative inverse?

The square of the integer

The constant of proportionality in a proportional relationship represents the ______ rate between two variables.

<p>unit</p>
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Match the following operations with their correct descriptions:

<p>Markup = Adding an amount to the cost of a product to determine the selling price. Markdown = Reducing the original selling price of a product. Simple Interest = Interest calculated only on the principal amount. Percent Error = The difference between an estimated value and an actual value, expressed as a percentage.</p>
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Which expression is equivalent to $2(x + 3) - (x - 1)$?

<p>$x + 7$ (D)</p>
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A percent increase of 50% followed by a percent decrease of 50% will result in the original value.

<p>False (B)</p>
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If two ratios are equivalent, what term describes the relationship between them?

<p>Proportional</p>
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When solving for simple interest, the formula used is I = P * r * t, where 'I' represents interest, 'P' represents principal, 'r' represents the interest rate, and 't' represents ______.

<p>time</p>
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Match each term related to algebraic expressions with its description:

<p>Factor = To rewrite an expression as a product of its components. Expand = To multiply out terms in an expression, often using the distributive property. Simplify = To rewrite an expression in its most compact form. Evaluate = To find the value of an expression by substituting given values for variables.</p>
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What does the graph of a proportional relationship always include?

<p>A straight line that passes through the origin (A)</p>
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Dividing two negative integers always results in a negative quotient.

<p>False (B)</p>
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Define 'unit rate' in the context of ratios and proportional relationships.

<p>A rate with a denominator of 1</p>
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An expression that can be written as a sum of terms, where each term includes a constant and variable raised to a whole number exponent, is called an ______ expression.

<p>algebraic</p>
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Match the operation with its effect on an algebraic expression:

<p>Adding Like Terms = Simplifies the expression by combining terms with the same variable and exponent. Distributing = Expands the expression by multiplying a term across the terms inside parentheses. Factoring = Rewrites the expression by identifying common factors within the terms. Evaluating = Calculates the numerical value of the expression for given values of the variables.</p>
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Which of the following situations involves calculating a percent error?

<p>Comparing an estimated measurement to the actual measurement. (C)</p>
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The reciprocal of a rational number is always another rational number.

<p>True (A)</p>
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How is the constant of proportionality related to the unit rate in a proportional relationship?

<p>It equals the unit rate</p>
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In the equation $y = kx$, 'k' represents the ______ of proportionality between 'x' and 'y'.

<p>constant</p>
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Match the algebraic property with its correct description:

<p>Commutative Property = The order of operations does not affect the result (e.g., a + b = b + a). Associative Property = The grouping of operations does not affect the result (e.g., (a + b) + c = a + (b + c)). Distributive Property = Multiplying a sum by a number is the same as multiplying each addend separately and then adding the products (e.g., a(b + c) = ab + ac). Identity Property = A number remains unchanged when added to zero or multiplied by one.</p>
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Flashcards

Rational Number

A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Absolute Value

The distance of a number from zero on the number line, always a non-negative value.

Integer Opposites

Two numbers that are the same distance from zero on the number line but in opposite directions.

Ratio

A comparison of two quantities by division.

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Unit Rate

A rate that compares a quantity to one unit of another quantity.

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Proportion

An equation stating that two ratios are equal.

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Proportional Relationship

Relationships where two quantities vary directly with each other; If one changes, the other changes by a constant factor.

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Constant of Proportionality

The constant value of the ratio between two proportional quantities, often denoted as 'k'.

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Percent Change

A ratio that compares a change in quantity to the original amount, expressed as a percentage.

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Percent Error

The difference between an estimated value and the actual value, expressed as a percentage of the actual value.

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Markup

Adding a percentage to the cost of a product to make a profit.

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Markdown

Reducing the selling price of a product, usually expressed as a percentage of the original price.

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Simple Interest

A fee paid or received for the use of money.

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Algebraic Expression

A combination of variables, numbers, and operations.

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Equivalent Expressions

Expressions that have the same value for all possible values of the variables.

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Simplify Expressions

To rewrite an expression in a simpler form by combining like terms.

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Expand Expressions

Applies the distributive property to remove parentheses.

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Factor Expressions

Rewriting an expression as the product of its factors.

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Study Notes

  • Integers and their opposites are related concepts in mathematics.
  • Rational numbers encompass integers and can be subjected to arithmetic operations.
  • Adding integers is a fundamental operation with specific rules depending on the signs of the integers.
  • Subtracting integers involves adding the opposite of the integer being subtracted.
  • Rational numbers can also be added and subtracted, following rules for fractions and decimals.
  • Multiplying integers requires consideration of the signs involved to determine the sign of the product.
  • Multiplying rational numbers involves multiplying fractions or decimals.
  • Dividing integers also requires attention to signs to determine the sign of the quotient.
  • Dividing rational numbers involves dividing fractions or decimals.
  • Problem-solving with rational numbers applies these arithmetic operations in various contexts.
  • Ratios, rates, and unit rates are connected concepts used to compare quantities.
  • Unit rates can be determined from ratios of fractions.
  • Proportional relationships involve a constant ratio between two quantities.
  • Equivalent ratios represent the same proportional relationship.
  • Proportional relationships can be described using equations or words.
  • Constant proportionality is the constant value of the ratio in a proportional relationship.
  • Proportional relationships can be represented graphically as a straight line through the origin.
  • Proportional reasoning is applied to solve problems involving proportional relationships.
  • Percents are related to ratios and proportions.
  • Percent can be represented as a proportion.
  • The percent equation is used to solve percent problems: Part = Percent × Whole.
  • Percent of change and percent error problems involve calculating the relative difference between two values.
  • Markup and markdown problems are applications of percent change in retail settings.
  • Simple interest problems involve calculating interest earned or paid on a principal amount.
  • Algebraic expressions can be written and evaluated.
  • Equivalent expressions have the same value for all values of the variables.
  • Expressions can be simplified by combining like terms and using the order of operations.
  • Expanding expressions involves removing parentheses by applying the distributive property.
  • Factoring expressions involves writing an expression as a product of factors.
  • Expressions can be added and subtracted by combining like terms.
  • Analyzing equivalent expressions involves determining if two expressions are equivalent.

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