Decimals, Fractions, and Percentages

• These study notes cover decimals, fractions, and percentages
• Rounding decimals is a key focus

Rounding Decimals

• To round decimals, identify the decimal place to which rounding is needed
• If the next digit is 5 or more, round up; otherwise, keep the digit the same
• For example, rounding 34.648 to one decimal place results in 34.6
• Rounding 12.7643 to three decimal places gets 12.764
• Rounding 189.795 cents to two decimal places makes it 189.80 cents
• 189.795 cents becomes $1.90 when rounded to the nearest dollar

Adding and Subtracting Decimals

• Add and subtract decimals by aligning the decimal points
• Carry out the operations as you would with whole numbers
• Ensure to place the decimal point in the correct position in the answer

Missing Numbers in Sums

• Work backwards using addition or subtraction to find missing numbers in decimal sums
• For example, solve for missing digits by understanding relationships

Word Problems with Decimals

• To solve word problems, identify the operations needed
• Extract relevant information and apply the correct steps
• For example, find the remaining distance in a race by subtracting the distances

Multiplying and Dividing Decimals by 10, 100, 1000

• Multiplying by 10, 100, or 1000 involves shifting the decimal point to the right
• Move the decimal one place for each zero in the multiplier
• Dividing by 10, 100, or 1000 involves shifting the decimal point to the left

Estimating Products

• Estimate products by rounding decimals to the nearest whole number
• Multiply the rounded numbers to approximate the actual product

Calculating Products

• Multiply decimals like whole numbers
• Count the total number of decimal places in the factors
• Place the decimal point in the product so it has the same number of decimal places

Using Given Products

• Apply known products to deduce results for related decimal multiplications
• For instance, if 256 x 1289 = 329984, then 25.6 x 1289 = 32998.4

Dividing Decimals

• Divide decimals using long division
• Ensure the decimal point is correctly placed in the quotient
• Dividing 25.968 by 4 equals 6.492

More Decimal Division

• Divide decimals by decimals by multiplying both by powers of 10 to get rid of the decimal in the divisor
• Dividing 1.4632 by 0.04 results in 36.58

Real-World Decimal Problems

• Solve practical problems involving decimals using appropriate operations
• This may include division to find the price per unit

Equivalent Fractions

• Fractions represent parts of a whole
• Equivalent fractions denote the same portion of a whole
• Numerator/denominator combinations include one whole (1/1), halves (1/2), thirds (1/3), quarters (1/4), fifths (1/5)

Simplifying Fractions

• Simplify fractions by dividing both the numerator and the denominator
• Reducing 2/10 gives 1/5

Comparing Fractions

• To compare fractions, find a common denominator
• Compare the numerators
• For example: Compare 10/13 and 11/13, 11/13 is greater.

Expanding Fractions

• To expand fractions, multiply both the numerator and denominator
• This gives them a common denominator for comparison or addition.

Improper and Mixed Fractions

• Improper fractions have a numerator larger than the denominator
• Mixed fractions include a whole number and a proper fraction (e.g., 2 1/3)
• To convert an improper fraction to a mixed fraction, divide the numerator by the denominator
• Convert a mixed fraction to an improper fraction, mulitply the whole number and add to the numerator.

Ordering Fractions

• To order fractions place the math symbol of <, =, or > between them

Ascending Order

• Place a group of fractrions in ascending order (smallest to largest)

Adding Fractions

• Add fractions with a common denominator by adding the numerators and keeping the denominator
• To add fractions with different denominators, find a common denominator and add the adjusted numerators

Subtracting Fractions

• Subtract fractions with a common denominator by subtracting the numerators
• For those with different denominators, find a common denominator first

Multiplying Fractions

• Multiply fractions by multiplying the numerators and the denominators separately
• Simplify the result of multiplying the numerators and denominators if necessary

Dividing Fractions

• Dividing fractions involves multiplying by the reciprocal of the divisor
• Multiply by the reciprocal after replacing the ÷ sign to X

Introduction to Ratios

• A ratio compares amounts or quantities
• Ratios can compare parts to parts or parts to a whole

Formulating Ratios

• Formulate ratios for different real world items

Simplifying Ratios

• Simplify ratios by dividing all parts by their greatest common factor
• You can do this if all sides are divisible, for example 15:35 can be 3:5

Solving Ratio Problems

• Use ratios to solve problems
• Setup the ratios as fractions

Juice Concentrate

• You can find the ratio of what ammount of juice concentrate to water should be mixed and solve

Fractions and Percentages

• A percentage can be converted into a fraction by expressing it with 100 as the denominator
• 7% is 7/100

Percentages to Fractions

• To convert a percentage into a fraction, use 100 as denominator then simplified

Fractions to Percentages

• To convert fractions to percentages, multiply the fraction by 100%

Percentages of Tart

• Real world percentages, a lemon tart cut into 8 equal pieces would mean each piece is 12.5%

Percentage to Decimal

• A percentage can be converted into a decimal by removing the ‘%’ sign and divide by 100
• 45.6% is .456

Decimal to Percentage

• A decimal can be converted into a percentage by multiplying the desimal by 100 and adding the "%" symbol

Finding a Percentage of a Number

• To find p% of n, calculate n × p%

Office Employees

• You can discover real world scenarios using percentages of a number to show what the actual amount is