Decimals
52 flashcards covering Decimals for the GMAT Quantitative section.
Decimals are a straightforward way to represent numbers that include fractional parts, such as 0.75 or 3.14. They extend the base-10 system we use for whole numbers, allowing us to express values between integers with precision. For instance, decimals help describe measurements, money, or percentages in everyday life, making them essential for accurate calculations and comparisons.
On the GMAT Quantitative section, decimals often appear in arithmetic problems, algebra, and data interpretation questions. You'll encounter tasks like adding, subtracting, multiplying, or dividing decimals, as well as converting them to fractions or percentages. Common traps include misplaced decimal points, rounding errors, and overlooking place values, which can lead to incorrect answers. Focus on practicing operations with decimals in word problems and double-checking your work to build speed and accuracy.
Remember to convert decimals to fractions when dealing with ratios for clearer insights.
Terms (52)
- 01
What is a decimal?
A decimal is a number that expresses a quantity less than one or greater than a whole number using a decimal point, such as 0.5 or 3.14, to separate the integer part from the fractional part.
- 02
Place value in decimals
In decimals, place value refers to the value of each digit based on its position relative to the decimal point, where each place to the right represents a power of ten, like tenths, hundredths, and thousandths.
- 03
Adding decimals
Adding decimals involves aligning the decimal points of the numbers and then adding them digit by digit, just like whole numbers, to ensure accuracy in the fractional parts.
- 04
Subtracting decimals
Subtracting decimals requires aligning the decimal points and subtracting digit by digit, often adding zeros to make the numbers align properly without changing their value.
- 05
Multiplying decimals
Multiplying decimals means first multiplying the numbers as if they were whole numbers and then placing the decimal point in the product so that the total number of decimal places equals the sum of those in the factors.
- 06
Dividing decimals
Dividing decimals involves making the divisor a whole number by multiplying both dividend and divisor by the same power of ten, then performing the division as with whole numbers.
- 07
Converting decimal to fraction
Converting a decimal to a fraction means expressing the decimal as a fraction by placing the digits after the decimal point over a denominator that is a power of ten, then simplifying if possible.
- 08
Converting fraction to decimal
Converting a fraction to a decimal involves dividing the numerator by the denominator, which may result in a terminating, repeating, or non-repeating decimal depending on the fraction.
- 09
Rounding decimals
Rounding decimals means adjusting a number to a specified place value by looking at the digit immediately to the right; if it's 5 or greater, increase the target digit by one.
- 10
Comparing decimals
Comparing decimals requires aligning the decimal points and comparing digits from left to right; if one has more digits, add zeros to the end of the other for accurate comparison.
- 11
Decimals on the number line
Decimals on the number line are placed according to their value, with positive decimals to the right of zero and negative decimals to the left, allowing for visualization of their relative sizes.
- 12
Multiplying by powers of 10
Multiplying a decimal by a power of 10, like 10 or 100, moves the decimal point to the right by the number of zeros in that power, effectively increasing the number's magnitude.
- 13
Dividing by powers of 10
Dividing a decimal by a power of 10 moves the decimal point to the left by the number of zeros, which decreases the number's magnitude without changing its value otherwise.
- 14
Scientific notation for decimals
Scientific notation expresses decimals as a number between 1 and 10 multiplied by a power of 10, useful for very large or small numbers, like 5.67 times 10 to the power of 3.
- 15
Terminating decimals
Terminating decimals are those that end after a finite number of digits after the decimal point, such as 0.25, and can be exactly represented as fractions.
- 16
Repeating decimals
Repeating decimals have a sequence of digits that repeats indefinitely, like 0.333..., and can be converted to fractions for exact representation.
- 17
Negative decimals
Negative decimals represent values less than zero, such as -2.5, and follow the same rules for operations as positive decimals but result in negative outcomes where appropriate.
- 18
Comparing positive and negative decimals
Comparing positive and negative decimals involves recognizing that any positive decimal is greater than any negative decimal, and among negatives, the one closer to zero is larger.
- 19
Decimals in percentages
Decimals in percentages are used to express parts per hundred, like 0.05 meaning 5%, and can be converted by multiplying the decimal by 100 and adding a percent sign.
- 20
Percent increase with decimals
Percent increase with decimals calculates the growth relative to the original amount, such as finding a 12.5% increase on 50 by multiplying 50 by 0.125 and adding the result.
- 21
Ratios with decimals
Ratios with decimals express proportional relationships, like 1.5:2, and can be simplified by multiplying both parts by the same number to eliminate decimals.
- 22
Solving equations with decimals
Solving equations with decimals involves isolating the variable using standard algebraic operations, often by multiplying through by a power of ten to eliminate decimals first.
- 23
Inequalities with decimals
Inequalities with decimals follow the same rules as those with whole numbers, but care must be taken with multiplication or division by negative numbers to reverse the inequality sign.
- 24
Absolute value of decimals
The absolute value of a decimal is its distance from zero on the number line, so for example, the absolute value of -3.2 is 3.2, ignoring the sign.
- 25
Order of operations with decimals
Order of operations with decimals follows PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), treating decimals like any other numbers in the sequence.
- 26
Exponents with decimal bases
Exponents with decimal bases mean raising the decimal to a power, like (0.5)^2 = 0.25, which requires multiplying the base by itself the specified number of times.
- 27
Roots of decimal numbers
Roots of decimal numbers, such as the square root of 1.44 which is 1.2, involve finding a number that, when multiplied by itself, equals the original decimal.
- 28
Common mistake in adding decimals
A common mistake in adding decimals is misaligning the decimal points, which can be avoided by lining them up vertically before performing the addition.
- 29
Common mistake in multiplying decimals
A common mistake in multiplying decimals is forgetting to count the total decimal places in the factors for the product, leading to incorrect placement of the decimal point.
- 30
Strategy for estimating with decimals
A strategy for estimating with decimals is to round the numbers to the nearest whole number or easy fraction first, then perform the operation to get a quick approximate answer.
- 31
Converting mixed numbers to decimals
Converting mixed numbers to decimals involves converting the fractional part to a decimal and adding it to the whole number, like 2 and 1/2 becomes 2.5.
- 32
Decimals in weighted averages
Decimals in weighted averages are used when the weights are not equal, requiring multiplication of each value by its weight, summing the products, and dividing by the total weight.
- 33
Probability with decimal outcomes
Probability with decimal outcomes expresses the likelihood of an event as a decimal between 0 and 1, where 0 means impossible and 1 means certain, calculated as favorable outcomes over total outcomes.
- 34
Interest calculations with decimals
Interest calculations with decimals, such as simple interest, use formulas like I = P R T, where rate R might be a decimal like 0.05 for 5%, to find the interest earned.
- 35
Discount calculations with decimals
Discount calculations with decimals involve multiplying the original price by the discount rate, such as 0.20 for 20%, and subtracting the result from the original price.
- 36
Decimals in geometric formulas
Decimals in geometric formulas, like area of a circle A = πr², allow for precise calculations when the radius is a decimal, such as 3.14 times (2.5)^2.
- 37
Worked example: Adding 1.23 and 4.56
Adding 1.23 and 4.56 gives 5.79, by aligning the decimals and adding column by column.
- 38
Worked example: Multiplying 2.5 by 3.4
Multiplying 2.5 by 3.4 first gives 8.5 as the product of 25 and 34 divided by 100, since there are two decimal places total.
- 39
Worked example: Dividing 10 by 0.5
Dividing 10 by 0.5 equals 20, because multiplying both by 10 makes it 100 divided by 5, which is 20.
- 40
Trap: Forgetting zeros in division
A trap in decimal division is forgetting to add zeros to the dividend if needed, which can lead to incorrect results, like dividing 1 by 0.1 without considering it equals 10.
- 41
Trap: Misplacing decimal in multiplication
Misplacing the decimal in multiplication can result from not counting the decimal places correctly, such as in 0.2 times 0.3, which should be 0.06, not 0.6.
- 42
Decimal equivalents of common fractions
Decimal equivalents of common fractions are useful shortcuts, like 1/2 = 0.5, 1/4 = 0.25, and 3/4 = 0.75, for quick conversions during calculations.
- 43
Long division with decimals
Long division with decimals follows the standard process but requires moving the decimal point in the dividend to make the divisor a whole number, then adjusting the decimal in the quotient.
- 44
Irrational decimals
Irrational decimals, like the decimal form of pi (3.14159...), go on forever without repeating and cannot be expressed as simple fractions.
- 45
Truncating decimals
Truncating decimals means cutting off the decimal at a certain point without rounding, such as truncating 3.14159 to 3.14, which differs from rounding.
- 46
Decimals in fractions greater than 1
Decimals in fractions greater than 1, like 2.5 which is 5/2, represent mixed numbers and can be used in operations just like other decimals.
- 47
Adding decimals with different places
Adding decimals with different decimal places requires aligning the points and adding zeros to the shorter one, ensuring all columns are accounted for.
- 48
Multiplying decimals by whole numbers
Multiplying decimals by whole numbers involves treating the decimal as usual and placing the decimal point in the product based on the original decimal places.
- 49
Decimal division by whole numbers
Decimal division by whole numbers, like 10 divided by 2 equals 5, follows standard division rules, with the quotient potentially being a decimal if not exact.
- 50
Strategy for decimal word problems
A strategy for decimal word problems is to identify the operation needed, convert units if necessary, perform the calculation accurately, and check the context for reasonable results.
- 51
Decimals in data interpretation
Decimals in data interpretation appear in charts or tables, requiring careful reading and calculation, such as averaging decimal values from a graph.
- 52
Estimating products of decimals
Estimating products of decimals involves rounding to whole numbers first, multiplying, and then adjusting, to quickly check if an exact answer is reasonable.