Algebra Class: Ratios and Number Concepts

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Questions and Answers

What is the definition of a ratio?

A comparison of two quantities, expressed as a:b or a/b.

How do you simplify a ratio?

Divide both terms by their greatest common factor.

What is a proportion?

An equation stating two ratios are equal, e.g., a/b = c/d.

How do you solve a proportion?

<p>Cross-multiply.</p> Signup and view all the answers

What is a rate?

<p>A ratio comparing quantities of different units, e.g., speed = distance/time.</p> Signup and view all the answers

Select all of the following that are integers:

<p>7 (B), 0 (C), -5 (D)</p> Signup and view all the answers

What is the prime factorization of 12?

<p>2 x 2 x 3</p> Signup and view all the answers

What is the LCM of 8 and 12?

<p>24</p> Signup and view all the answers

Round 3.14159 to two decimal places.

<p>3.14</p> Signup and view all the answers

Simplify the expression 2x + 3y - 5x + 7y.

<p>-3x + 10y</p> Signup and view all the answers

Solve the equation 2x + 5 = 11 for x.

<p>x = 3</p> Signup and view all the answers

Represent the inequality x < 5 on a number line.

<p>A number line with an open circle at 5 and everything to the left of the circle shaded.</p> Signup and view all the answers

Given x = 2 and y = 3, find the value of the expression 3x + 2y.

<p>12</p> Signup and view all the answers

Simplify the fraction 12/18.

<p>2/3</p> Signup and view all the answers

Convert the improper fraction 7/4 into a mixed number.

<p>1 3/4</p> Signup and view all the answers

Express the number 3,200,000 in standard form.

<p>3.2 x 10^6</p> Signup and view all the answers

Flashcards

Ratio

A comparison of two quantities, expressed as a:b or a/b.

Simplify a ratio

Divide both parts of the ratio by their greatest common factor (GCF).

Proportion

An equation stating that two ratios are equal, like a/b = c/d.

Solve a proportion

Use cross-multiplication to find the unknown value.

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Rate

A ratio comparing quantities with different units (e.g., speed).

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Integer

A whole number (positive, negative, or zero).

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Fraction

A number representing part of a whole.

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Decimal

A number with a decimal point.

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Percentage

A fraction or ratio expressed out of 100.

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Prime Factorization

Breaking a number down into its prime factors.

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LCM

Least Common Multiple

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HCF

Highest Common Factor

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Expression

A combination of variables and numbers using operations.

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Combine like terms

Add or subtract terms with the same variables.

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Equation

A statement that two expressions are equal.

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Inequality

A statement that two expressions are not equal.

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Substitution

Replacing a variable with a value in an expression.

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Improper Fraction

A fraction where the numerator is greater than or equal to the denominator.

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Mixed Number

A number that has a whole number and a fraction part.

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Standard Form

Expressing a number as a×10n, where 1 ≤ a < 10.

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Study Notes

Ratios

  • A ratio compares two quantities.
  • It can be expressed as a fraction, using a colon, or with the word "to".
  • Example: The ratio of 3 apples to 5 oranges can be written as 3/5, 3:5, or 3 to 5.

Simplifying Ratios

  • To simplify a ratio, divide both terms by their greatest common factor (GCF).
  • Example: The ratio 6:8 can be simplified to 3:4 by dividing both terms by 2.

Proportions

  • A proportion is an equality of two ratios.
  • It states that two ratios are equivalent.
  • Example: 2/3 = 4/6 is a proportion because the two ratios are equivalent.

Solving Proportions

  • To solve a proportion, cross-multiply and then solve for the unknown variable.
  • Example: To solve the proportion 2/x = 3/5, cross-multiply to get 2 * 5 = 3 * x, then solve for x to get x = 10/3.

Rates

  • A rate is a ratio that compares two quantities with different units.
  • Example: A speed of 60 miles per hour is a rate that compares distance (miles) to time (hours).

Integers

  • Integers are whole numbers that can be positive, negative, or zero.
  • Example: -5, 0, 3, 10 are all integers.

Prime Factorization of 12

  • The prime factorization of 12 is 2 x 2 x 3.

Least Common Multiple (LCM) of 8 and 12

  • The LCM of 8 and 12 is 24.

Rounding 3.14159 to Two Decimal Places

  • 3.14159 rounded to two decimal places is 3.14.

Simplifying the Expression 2x + 3y - 5x + 7y

  • The simplified expression is -3x + 10y.

Solving the Equation 2x + 5 = 11 for x

  • Solving for x:
    • Subtract 5 from both sides: 2x = 6
    • Divide both sides by 2: x = 3.

Representing the Inequality x < 5 on a Number Line

  • The number line will have an open circle at 5 and an arrow pointing to the left, indicating all numbers less than 5.

Evaluating the Expression 3x + 2y

  • Given x = 2 and y = 3:
    • Substitute the values: 3(2) + 2(3)
    • Evaluate: 6 + 6 = 12.

Simplifying the Fraction 12/18

  • The simplified fraction is 2/3.

Converting the Improper Fraction 7/4 into a Mixed Number

  • The mixed number is 1 3/4.

Expressing 3,200,000 in Standard Form

  • The standard form is 3.2 x 10^6.

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