Fractions and decimals
59 flashcards covering Fractions and decimals for the ACT Math section.
Fractions represent parts of a whole, like dividing a pizza into equal slices where 1/2 means one slice out of two. Decimals express these parts using a decimal point, such as 0.5 for half. Together, they form the foundation of many math concepts, allowing us to handle measurements, percentages, and ratios in everyday life and more advanced problems.
On the ACT Math section, fractions and decimals appear in questions involving operations like addition or multiplication, conversions between the two forms, and word problems that require real-world application. Common traps include forgetting to simplify fractions or mishandling decimal places, which can lead to incorrect answers. Focus on mastering equivalences, such as knowing that 3/4 equals 0.75, and practicing quick mental math to save time during the test.
A helpful tip: Always double-check your conversions to avoid simple errors.
Terms (59)
- 01
Fraction
A fraction is a numerical quantity that represents a part of a whole, expressed as one number divided by another, such as 1/2.
- 02
Numerator
The numerator is the top number in a fraction, indicating how many parts of the whole are being considered.
- 03
Denominator
The denominator is the bottom number in a fraction, indicating the total number of equal parts the whole is divided into.
- 04
Proper fraction
A proper fraction is a fraction where the numerator is less than the denominator, such as 3/4, representing a value less than one.
- 05
Improper fraction
An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as 5/3, representing a value greater than or equal to one.
- 06
Mixed number
A mixed number consists of a whole number plus a proper fraction, such as 2 1/2, used to express values greater than one.
- 07
Equivalent fractions
Equivalent fractions are different fractions that represent the same value, such as 1/2 and 2/4, found by multiplying or dividing both numerator and denominator by the same number.
- 08
Simplifying fractions
Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and denominator by their greatest common divisor.
- 09
Greatest common divisor
The greatest common divisor of two numbers is the largest number that divides both without leaving a remainder, used in simplifying fractions.
- 10
Least common denominator
The least common denominator is the smallest number that is a multiple of two or more denominators, used when adding or subtracting fractions.
- 11
Adding fractions
Adding fractions requires a common denominator; add the numerators and keep the denominator the same, then simplify if possible.
- 12
Subtracting fractions
Subtracting fractions requires a common denominator; subtract the numerators and keep the denominator the same, then simplify if possible.
- 13
Multiplying fractions
Multiplying fractions involves multiplying the numerators together and the denominators together, then simplifying the result.
- 14
Dividing fractions
Dividing fractions requires multiplying the first fraction by the reciprocal of the second, then simplifying the result.
- 15
Reciprocal of a fraction
The reciprocal of a fraction is obtained by swapping its numerator and denominator, such as the reciprocal of 2/3 being 3/2.
- 16
Decimal
A decimal is a number expressed in base ten using a decimal point to separate the whole number part from the fractional part, such as 0.5.
- 17
Terminating decimal
A terminating decimal is a decimal that ends after a finite number of digits, such as 0.75.
- 18
Repeating decimal
A repeating decimal is a decimal with a sequence of digits that repeats indefinitely, such as 0.333... where the 3 repeats.
- 19
Converting fraction to decimal
Converting a fraction to a decimal involves dividing the numerator by the denominator, resulting in either a terminating or repeating decimal.
- 20
Converting decimal to fraction
Converting a decimal to a fraction means expressing the decimal as a fraction in its simplest form, such as 0.5 becoming 1/2.
- 21
Place value in decimals
Place value in decimals refers to the value of each digit based on its position, such as the 5 in 0.5 representing five tenths.
- 22
Comparing fractions
Comparing fractions involves finding a common denominator or converting to decimals to determine which is larger or smaller.
- 23
Comparing decimals
Comparing decimals requires aligning the decimal points and comparing digits from left to right to determine which is larger or smaller.
- 24
Ordering fractions and decimals
Ordering fractions and decimals means arranging them from least to greatest or vice versa by converting them to a common form, like all decimals.
- 25
Multiplying decimals
Multiplying decimals involves multiplying the numbers as if they were whole numbers and then placing the decimal point in the product to match the total decimal places in the factors.
- 26
Dividing decimals
Dividing decimals requires eliminating decimals by multiplying both dividend and divisor by a power of ten, then performing the division as with whole numbers.
- 27
Rounding decimals
Rounding decimals means adjusting a decimal to a specified number of places by looking at the digit immediately after and following standard rounding rules.
- 28
Fraction of a number
A fraction of a number is found by multiplying the number by the fraction, such as finding one-half of 10 by calculating 10 times 1/2.
- 29
Percentage as a fraction
A percentage is expressed as a fraction with a denominator of 100, such as 25% being equivalent to 25/100 or 1/4.
- 30
Ratio and fractions
A ratio can be expressed as a fraction, representing the relationship between two quantities, such as 2:3 being the same as 2/3.
- 31
Complex fraction
A complex fraction is a fraction where the numerator, denominator, or both are themselves fractions, simplified by multiplying by the reciprocal or finding a common denominator.
- 32
Adding mixed numbers
Adding mixed numbers involves adding the whole numbers separately and the fractions separately after finding a common denominator, then combining if needed.
- 33
Subtracting mixed numbers
Subtracting mixed numbers may require borrowing from the whole number if the fractional parts don't subtract directly, after finding a common denominator.
- 34
Proportions
Proportions are equations where two ratios or fractions are equal, such as a/b = c/d, solved using cross-multiplication.
- 35
Cross-multiplication
Cross-multiplication is a method to solve proportions by multiplying the numerator of one fraction by the denominator of the other and setting them equal.
- 36
Solving proportions with fractions
Solving proportions with fractions involves cross-multiplying and then solving for the unknown variable.
- 37
Benchmark fractions
Benchmark fractions are common fractions like 1/2, 1/3, and 2/3 used as reference points for estimating and comparing other fractions.
- 38
Fraction-decimal-percent equivalents
Fraction-decimal-percent equivalents are the interchangeable forms of the same value, such as 1/2 equaling 0.5 and 50%.
- 39
Operations with negative fractions
Operations with negative fractions follow the same rules as positive ones but account for the signs, such as negative times negative yielding positive.
- 40
Operations with negative decimals
Operations with negative decimals involve applying the rules of signs, like adding two negatives results in a more negative number.
- 41
Least common multiple
The least common multiple of two numbers is the smallest number that is a multiple of both, often used with fractions for common denominators.
- 42
Fraction bars in equations
Fraction bars in equations act as grouping symbols, indicating that the numerator and denominator should be treated as a single unit.
- 43
Decimals in inequalities
Decimals in inequalities are compared and manipulated using the same rules as numbers, preserving the inequality sign during operations.
- 44
Word problems involving fractions
Word problems involving fractions require translating phrases into fractional equations and solving them step by step.
- 45
Word problems involving decimals
Word problems involving decimals involve setting up equations with decimals based on the problem's context and performing the necessary operations.
- 46
Rates and fractions
Rates are ratios expressed as fractions, such as speed as distance over time, and can be simplified or compared like fractions.
- 47
Unit rates
Unit rates are rates expressed as a fraction with a denominator of one, found by dividing the numerator by the denominator.
- 48
Common mistakes in fraction addition
A common mistake in fraction addition is adding numerators and denominators separately without a common denominator, leading to incorrect results.
- 49
Common mistakes in decimal division
A common mistake in decimal division is forgetting to move the decimal point in both the dividend and divisor, resulting in wrong quotients.
- 50
Scientific notation for decimals
Scientific notation expresses very large or small decimals as a number between 1 and 10 multiplied by a power of ten, such as 5.6 × 10^3 for 5600.
- 51
Example of simplifying 4/8
Simplifying 4/8 involves dividing both numerator and denominator by 4, resulting in 1/2.
divided by 4 is 1, and 8 divided by 4 is 2, so 4/8 simplifies to 1/2.
- 52
Example of adding 1/4 and 1/4
Adding 1/4 and 1/4 requires a common denominator, which they share, so 1/4 + 1/4 equals 2/4, simplifying to 1/2.
- 53
Example of multiplying 0.5 by 2
Multiplying 0.5 by 2 gives 1.0, showing how decimal multiplication works with whole numbers.
- 54
Example of converting 3/4 to decimal
Converting 3/4 to a decimal involves dividing 3 by 4, resulting in 0.75.
- 55
Example of dividing fractions 1/2 by 1/4
Dividing 1/2 by 1/4 means multiplying 1/2 by 4/1, resulting in 4/2 or 2.
- 56
Example of comparing 0.7 and 2/3
Comparing 0.7 and 2/3 involves converting 2/3 to about 0.666, so 0.7 is greater.
- 57
Example of a proportion 2/3 = 4/x
Solving the proportion 2/3 = 4/x using cross-multiplication gives 2x = 12, so x = 6.
- 58
Strategy for avoiding decimal errors
A strategy for avoiding decimal errors is to count decimal places carefully and align them when adding or subtracting.
- 59
Strategy for fraction word problems
A strategy for fraction word problems is to identify the whole, the parts, and set up the fraction based on the question's context.