Quantitative Reasoning Study Guide for the DAT

Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics that are used to compare quantities and establish relationships between them. Ratios and proportions are used in various fields, including finance, engineering, and science, as they provide a framework for making comparisons and solving problems.

Ratios

A ratio is a comparison of two quantities using division. It expresses how many times one quantity is contained within another. Ratios are typically written in the form $\frac{a}{b}$, where $a$ and $b$ are numbers representing the quantities being compared. There are several ways to express a ratio:

fraction form—As discussed, ratios can be represented as fractions, such as $\frac{a}{b}$.
colon notation—Ratios can also be expressed using a colon, such as $a:b$.
word form—Ratios can be written in words, such as “$a$ to $b$” or “$a$ for every $b$”.

If the ratio of boys to girls in a classroom is $2$ to $3$, there are two boys for every three girls. We can write this ratio as $2:3$ or $\frac{2}{3}$.

Proportions

A proportion is an equation that states two ratios are equal. It represents a relationship of equivalence between two ratios. Unlike ratios, which simply compare quantities, proportions assert that two ratios have the same value. Proportions can be used to solve problems involving unknown quantities or for establishing relationships between known quantities.

For example, $\frac{a}{b} = \frac{c}{d}$ is a proportion where $a$, $b$, $c$, and $d$ are quantities and the ratios $\frac{a}{b}$ and $\frac{c}{d}$ are equal.

Using Proportions to Solve Problems

Proportions are powerful tools for solving various real-world problems, particularly those involving unknown quantities or proportional relationships. Here’s how to use proportions to solve problems: 1.

Identify the proportional relationship.—Determine which quantities are directly proportional to each other.
Set up the proportion.—Convert the given ratios to fractions and set them equal to each other.
Solve for the unknown.—Cross-multiply and solve the resulting equation to find the value of the unknown quantity.

Let’s try a problem.

A recipe calls for two cups of flour to make $24$ cookies. If you want to make $36$ cookies, how many cups of flour do you need?

First, we identify the proportional relationship. In this case, the amount of flour is directly proportional to the number of cookies. Now, set up the proportion:

\[\frac{\text{cups of flour}}{\text{number of cookies}} = \frac{2}{24} = \frac{x}{36}\]

Now, we solve for the unknown, $x$:

\[24 \times x = 2 \times 36\] \[24x = 72\] \[x = \frac{72}{24}\] \[x = 3\]

To make $36$ cookies, we need $3$ cups of flour.

Decimals

Decimals allow us to express values that fall between whole numbers. They are crucial for various applications, including measurements, calculations, and representing parts of a whole.

Basics

Decimal numbers consist of three parts:

whole number part—The digits to the left of the decimal point represent the whole number part.
decimal point—The decimal point is a dot (.) that separates the whole number part from the fractional part.
fractional part—The digits to the right of the decimal point represent the fractional part.

In the decimal number $123.45$, $123$ is the whole number part and $45$ is the fractional part.

Place Value

Place value refers to the value of a digit based on its position in a number. In decimal numbers, each digit’s value is determined by its position relative to the decimal point. Understanding place value is essential for accurately interpreting and manipulating decimal numbers. Below, we show a partial place value chart for decimals, listing the common place values:

3 Decimal Place Value Chart NEW.png

In this chart, each position represents a power of $10$. The digits to the left of the decimal point represent whole numbers, with each position corresponding to a power of $10$ (e.g., ones, tens, hundreds). The digits to the right of the decimal point represent fractions of a whole, with each position representing a negative power of $10$ (e.g., tenths, hundredths, thousandths).

For example, in the decimal $23.98$, the digit $3$ is in the ones place and the digit $2$ is in the tens place. Similarly, the digit $9$ is in the tenths place and the digit $8$ is in the hundredths place.

Converting Fractional Numbers

Converting between fractions and decimals is necessary for mathematical calculations and understanding numerical relationships. You need to know how to convert in both directions.

Fraction to Decimal

To convert a fraction into a decimal, divide the numerator by the denominator. To convert $\frac{3}{4}$ into a decimal, simply divide $3$ by $4$:

\[\frac{3}{4} = 3 \div 4 = 0.75\]

If you don’t have a calculator, you can use long division instead.

Decimal to Fraction

Converting a decimal to a fraction is simple if you know the place values after the decimal. The numerator is the fractional part (the numbers after the decimal point), and the denominator is the place value of the final digit (the one farthest to the right). You may need to simplify the fraction. An example problem will make the process clear.

Convert $0.72$ into a fraction.

First, we write the fractional part of the decimal, $72$, as the numerator of the fraction. The denominator will be $100$ since the fractional part has two digits, meaning it ends at the hundredths place value.

So, $0.72$ can be written as $\frac{72}{100}$.

Now, we need to reduce this fraction to its lowest terms using the LCM:

\[\frac{72}{100} = \frac{72 \div 4}{100 \div 4} = \frac{18}{25}\]

Operations with Decimals

Performing operations such as addition, subtraction, multiplication, and division with decimals is a crucial skill in mathematics. Let’s break down each operation and explore how to work with decimals effectively.

Adding and Subtracting Decimals

Adding and subtracting decimals follow similar processes to adding and subtracting whole numbers, with the added step of aligning the decimal points. To add decimals, align the decimal points and add the digits column by column, starting from the right.

4 Adding Decimals NEW.png

In the above problem, we added $5$ and $2$ to get $7$, $2$ and $4$ to get $6$, and $3$ and $1$ to get $4$.

Note: If the numbers being added do not have the same number of digits behind the decimal point, you may need to add zeros to make sure the decimal points align.

Subtracting decimals also requires aligning the decimal points and subtracting the digits column by column.

5 Subtracting Decimals NEW.png

Starting from the right in the above problem, since we cannot subtract $8$ from $2$, we “borrowed” a $10$ from $4$ and then subtracted all the digits.

Multiplying Decimals

When multiplying decimals, ignore the decimal points at first. Multiply the numbers as if they were whole numbers, and then place the decimal point in the product so that the number of decimal places in the product is equal to the sum of the decimal places in the factors.

6 Multiplying Decimals NEW.png

In the above problem, there were two decimal places in the factors (one in each), so after we got our product, we moved the decimal place two numbers to the left.

Dividing Decimals

Dividing decimals involves moving the decimal points in the divisor and dividend to make the divisor a whole number, performing the division, and then placing the decimal point in the quotient as in the division of whole numbers.

For instance, to divide $3.025$ by $2.2$, we make the divisor $22$ by multiplying by $10$. Doing the same thing with the dividend, it becomes $30.25$. The long division is shown below:

7 Dividing Decimals NEW.png

Thus, $3.025 \div 2.2 = 1.375$.

Scientific Notation

Scientific notation is a compact way of expressing very large or very small numbers using powers of $10$. It is commonly used in scientific and engineering fields to represent numbers with many zeros or very small values.

In scientific notation, a number is expressed as the product of a coefficient (a number between $1$ and $10$) and a power of $10$. The power of $10$ indicates how many places the decimal point needs to be moved to get the original number. It follows the form $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent, representing the power of $10$.

To write a number in scientific notation, you follow the steps shown below:

Move the decimal point.—You need to create a coefficient ($a$) between $1$ and $10$.
Count the places moved.—The number of places the decimal point was moved is the exponent ($b$). Moving left means the exponent will be positive, and moving right means the exponent will be negative.
Write the exponential form.—The number will be expressed as $a \times 10^b$.

Let’s try a sample problem

Write the numbers $3\text{,}200\text{,}000$ and $0.000675$ in scientific notation.

For $3\text{,}200\text{,}000$ to be written in scientific notation, you must have a coefficient of $3.2$. Thus, you have moved the decimal point six places to the left. So, the number is $3.2 \times 10^{6}$.

For $0.000675$, make it $6.75$. Thus, you have moved the decimal point four places to the right (the exponent will be negative). So, the number is $6.75 \times 10^{-4}$.

Multiplying/Dividing in Scientific Notation

When you multiply two numbers in scientific notation, you multiply the coefficients and add the exponents of $10$:

\[(4 \times 10^{6}) \times (2 \times 10^{2})\] \[= (4 \times 2) \times 10^{6+2}\] \[= 8 \times 10^{8}\]

For division, you divide the coefficients and subtract the exponents of $10$:

\[(4 \times 10^{6}) \div (2 \times 10^{2})\] \[= (4 \div 2) \times 10^{6-2}\] \[= 2 \times 10^{4}\]

Rounding Numbers

Rounding numbers is the process of approximating a numerical value to a specified degree of accuracy. It involves adjusting the value to a nearby value that is easier to work with or understand.

In everyday situations, rounding can streamline the process and make it more manageable. However, in scenarios where precision is crucial, such as scientific research, engineering, or financial calculations, rounding can lead to a loss of accuracy and affect the reliability of analyses.

As such, determining whether to round must be done on a case-by-case basis. In general, if you only need an estimate, rounding is a useful shortcut.

How to Round Numbers

When rounding, the rule is to look at the digit immediately to the right of the digit to which you are rounding. Depending on this digit, the rounding process follows these rules:

If the digit to the right is $\boldsymbol{5}$ or greater, round the digit up.
If the digit to the right is less than $\boldsymbol{5}$, round the digit down.
All the numbers to the right of the rounded digit become $0$ or are dropped if they are to the right of the decimal point.

Let’s look at a couple examples.

Round $3.786$ to the nearest tenth.

The tenths place value is the first digit to the right of the decimal point. In this case, that’s $7$. The digit immediately to the right of the tenths place is $8$, which is greater than $5$, so we round up. So, $3.786$ rounded to the nearest tenth is $3.8$.

Round $123.478$ to the nearest hundred.

The hundreds digit is $1$. The digit to its right is $2$, which is less than $5$, so we keep $1$ as is. So, $123.478$ rounded to the nearest hundred is $100$.

Percents

Percents represents parts of a whole as fractions of $100$. The word percent means “per one hundred.” Percentages have various applications, including finance, statistics, and everyday calculations. They serve as a universal language for expressing proportions and comparisons. For example, when shopping, we often encounter discounts expressed as percentages, indicating the proportion of the original price we can save. Similarly, in statistics, percentages help us interpret data by providing insights into proportions and distributions.

Basics

Percents are denoted using this symbol: $\%$ They represent fractions with a denominator of $100$. For instance, $50\%$ represents half of a whole, or $\frac{50}{100}$, while $100\%$ represents the entire whole. Percents can be converted to fractions and decimals for easier manipulation and comparison in mathematical operations.

Conversions

Converting between percents, fractions, and decimals enables us to work with percentages in various contexts.

Percent to Fraction

To convert a percent to a fraction, we write the percent as a fraction with a denominator of $100$ and simplify, if possible. This conversion is useful for understanding proportions and relationships between different quantities.

To convert $70\%$ to a fraction, we will write $70$ with a denominator of $100$:

\[\frac{70}{100}\]

Now, we reduce the fraction to its lowest terms:

\[\frac{70 \div 10}{100 \div 10} = \frac{7}{10}\]

Note: This fraction represents seven parts out of $10$ parts of the whole.

Percent to Decimal

Converting a percent to a decimal involves dividing the percent by $100$. You can also do this by moving the decimal point in the number two places to the left.

To convert $40\%$ to a decimal, you drop the percentage symbol and divide $40$ by $100$:

\[40 \div 100 = 0.4\]

Note: This is the same as moving the decimal point two places to the left. Remember, any number has an invisible decimal point at the end.

Decimal to Percent

Converting a decimal to a percent entails multiplying the decimal by $100$. It is equivalent to moving the decimal point in the number two places to the right and then tagging on the percentage symbol. This conversion helps us express decimal values as proportions of the whole, making them easier to interpret.

To convert $0.53$ into a percentage, we multiply by $100$:

\[0.53 \times 100 = 53\]

Now, we add the percentage symbol at the end to get $53\%$.

Fraction to Percent

Converting a fraction to a percent involves dividing the numerator by the denominator and multiplying by $100$. To convert the fraction $\frac{3}{5}$ into a percent, we will first do division:

\[\frac{3}{5} = 3 \div 5 = 0.6\]

Now, we multiply by $100$:

\[0.6 \times 100 = 60\]

Lastly, add the percent symbol at the end to get $60\%$.

So, $\frac{3}{5} = 60\%$.

Percent Increase and Decrease

Percent increase and decrease measure the change in a quantity compared to its original value. These concepts help us when we analyze trends, make comparisons, and determine changes over time.

Percent increase ($P_i$) represents the growth of a quantity from its original value ($OV$) to its new value ($NV$). Percent decrease ($P_d$) represents the decline of a quantity from its original value to its new value. The formulas are shown below:

\[P_i = \frac{NV - OV}{OV} \times 100\] \[P_d = \frac{OV - NV}{OV} \times 100\]

For example, you go to a grocery store and see that an item marked $\$70$ today was worth $\$50$ yesterday. How do you find the percentage increase?

Note that the new value is $\$70$ and the original value (with which we are comparing the increase) is $\$50$. We will put the numbers into the formula and find the percentage increase:

\[P_i = \frac{NV - OV}{OV} \times 100\] \[P_i = \frac{70 - 50}{50} \times 100\] \[P_i = \frac{20}{50} \times 100\] \[P_i = 40\%\]

Note: For percentage decrease problems, you follow the same procedure with the percent decrease formula.