Math Study Guide for the SHSAT

Ratios and Proportions

In mathematics, ratios and proportions are used to compare quantities and establish relationships between them.

Ratios and Rates

The difference between ratios and rates lies in the types of quantities being compared: ratios compare similar quantities, while rates compare different quantities and often involve a change over time or another quantity.

Expressing Ratios

A ratio is a comparison of two quantities that tells how many times one quantity is contained in another. It can be written in three different forms:

fraction form—$\frac{a}{b}$

colon form—$a \text{:} b$

“to” form—$a$ to $b$

For example, if you have three red marbles and five blue marbles, the ratio of red marbles to blue marbles can be expressed as $3\text{:}5$ or $\frac{3}{5}$.

Expressing Rates

Rates involve a change in one quantity over a change in another quantity, often involving time. They are represented by a special kind of ratio that includes units, such as miles per hour or dollars per day. Complex fractions can be used to express rates, where the numerator represents the change in quantity and the denominator represents the change in time or another quantity. Look at this example.

A car travels a distance of $120$ miles in two hours. Use a complex fraction to express the rate or speed of the car, which is typically measured in miles per hour (mph).

The rate of the car’s speed can be calculated as follows:

\[s = \frac{t}{d}\]

where $s$ is speed, $t$ is time, and $d$ is distance.

Substituting the given values into the formula:

\[s= \frac{120\,\text{mi}}{2\,\text{hr}}\] \[s= 60\, \text{mph}\]

Proportions

In mathematics, a proportion is a relationship between two equal ratios. A proportion expresses that two ratios are equivalent, meaning their cross-products are equal. For example, in the proportion $\frac{a}{b}=\frac{c}{d}$, the cross products $a \times d$ and $b \times c$ are equal.

Expressing a Proportion

To express a proportion, you can set up an equation using the given ratios. For instance, if you have the ratios $\frac{3}{5}$ and $\frac{9}{15}$, you can express the proportion as $\frac{3}{5}=\frac{9}{15}$.

To verify that this is correct, let’s cross-multiply the fractions:

\[3 \times 15 = 45\] \[5 \times 9 = 45\]

Identifying a Proportional Relationship

Identifying a proportional relationship involves recognizing that two quantities vary in a way that can be represented by equivalent ratios. For example, if the ratio of the number of miles driven to the amount of gas used remains constant, it indicates the relationship between distance and gas consumption is proportional.

Finding the Constant of Proportionality

The constant of proportionality, or unit rate, is a special case of proportionality where one of the ratios equals $\bf{1}$. This represents the rate of change of one quantity in relation to another. For example, if a car travels $60$ miles in $1$ hour, the unit rate is $60$ miles per hour.

Tables

Time (hours)	Distance Traveled (miles)
1	50
2	100
3	150
4	200

Now, let’s find the constant of proportionality or unit rate from this table using the following steps:

Identify the two quantities—The two quantities in this table are time and distance traveled.
Identify the corresponding values—Each row in the table represents a pair of corresponding values. For example, in $1$ hour the car travels $50$ miles, in $2$ hours it travels $100$ miles, and so on.
Calculate the ratios—Calculate the ratio of distance traveled to time for each pair. For instance, at $1$ hour the ratio is $\frac{50}{1}=50$ miles per hour, at $2$ hours it’s $\frac{100}{2}=50$ miles per hour, and so forth.
Look for a constant ratio—Examining the calculated ratios, we observe that they are all the same: $50$ miles per hour.
Interpret the constant ratio—The constant ratio of $50$ miles per hour represents the constant of proportionality or unit rate for this scenario. We can interpret this to mean that for every hour of time, the car travels $50$ miles.

By analyzing the table and calculating the ratios, we can determine the constant of proportionality or unit rate, which helps us understand the relationship between time and distance traveled.

Graphs

Graphs provide a visual depiction of the constant of proportionality or unit rate, offering insight into the relationship between two quantities. To identify the constant of proportionality on a graph, attention is often directed to specific points, notably the origin $(0,0)$ and a point where one quantity changes by a fixed amount, typically denoted as $(1, r)$, where $r$ signifies the unit rate.

For instance, consider a graph illustrating the relationship between distance traveled ($y$-axis) and time taken ($x$-axis). If the graph intersects the origin and passes through another point, such as ($1, 35$), where $35$ represents the distance traveled and $1$ represents the unit of time, then the unit rate is $35$ units of distance per unit of time, such as $35$ miles per hour. This signifies a consistent relationship between distance and time, providing a visual illustration of the constant of proportionality.

1 Rate of Change Graph (replacement).png

The slope of the graphed line represents the rate of change, with each unit of horizontal change corresponding to a fixed amount of vertical change. In a graph where distance is plotted against time, a constant slope indicates a constant unit rate.

Equations

In equations, the constant of proportionality or unit rate can be identified by examining the coefficients or constants.

For example, in the equation $y=mx$, the coefficient $m$ represents the unit rate, as it indicates the amount of change in $y$ for each unit change in $x$.

Consider this equation:

\[y=2x\]

Here, the coefficient $2$ represents the unit rate. For each unit increase in $x$, $y$ increases by $2$ units.

Diagrams

Diagrams are another way to provide a visual representation of the constant of proportionality or unit rate, offering a clear depiction of the relationship between two quantities. Diagrams facilitate easy understanding and interpretation of proportional relationships. Examining diagrams allows us to observe how quantities change relative to each other, making it easier to identify patterns of the constant of proportionality.

In this diagram of quantity and price, it is easy to see the pattern:

2 Constant Proportionality Image (NEW).png

For every $10$ liters of paint, the price goes up $\$20$.

Verbal Descriptions

Verbal descriptions are another way in which proportions can be presented on the test. These descriptions convey the constant of proportionality or unit rate in a regular sentence, such as what you might see in a word problem. For instance, you may be given this as a problem:

For every hour of time, a car traveled a distance of $50$ miles. What is the car’s unit rate?

Problem Solving

Solving problems with proportions involves finding unknown quantities in proportional relationships. These problems often arise when comparing two or more quantities that have a consistent ratio or when dealing with situations where one quantity varies directly or inversely with another. By understanding the principles of proportions, you can solve a wide range of real-world and mathematical problems.

In the next sections, we’ll explore various problem-solving strategies and techniques to tackle proportion-related questions effectively. These strategies may involve setting up and solving proportions, using cross-multiplication, applying direct or inverse variation formulas, or using algebraic manipulation to solve for unknowns.

Multi-Step Problems

A multi-step problem is one in which two or more mathematical operations are required to arrive at a solution. At least one step in solving multi-step proportion problems will use the concept of proportionality.

Suppose we have a problem where we are told a pole is $10$ feet tall and we need to find the height of a pole right next to it. If we know the $10$-foot pole casts a shadow that is $15$ feet long and the other pole casts a shadow of $20$ feet, we can find the unknown height by setting up a proportion where $x$ stands in for the unknown height:

\[\frac{x}{20}=\frac{10}{15}\]

Now, we can cross-multiply to solve for $x$:

\[15 \times x = 20 \times 10\] \[15x = 200\] \[x=13.33\]

Therefore, the second pole is $13.33$ feet tall.

Percentage Problems

Percentages are simply proportions expressed as a part of $100$ (“percent” literally means “of one hundred”). Some proportion problems will involve percentages, and you will need to convert a given percent to a proportion, or vice versa. Let’s try an example problem.

There are $40$$ students in a class and $60\%$ are girls. If $25\%$ of the girls have blue eyes, what proportion of girls in the class have blue eyes?

This is an example of a multi-step problem. Since $60\%$ of the students are girls ($g$), we will need to first find $60\%$ of $40$ before determining how many have blue eyes ($b$).

The quickest way to do this is to convert the percentage to a decimal by dividing it by $100$ (moving the decimal point two places to the left):

\[60\% = \frac{60}{100} = .6\] \[g= .6 \times 40 = 24\]

There are $24$ girls in the class. Now, since we know $25\%$ of the girls have blue eyes, we find $25\%$ of $24$:

\[b= .25 \times 24 = 6\]

Now, to find the proportion of girls in the class who have blue eyes, we put that over the total number of students in the class and reduce as necessary:

\[\frac{6}{40} = \frac{3}{20}\]

So, the proportion of girls in the class with blue eyes is $\frac{3}{20}$.