Download Chapter 11.1 – Ratios and Rates A ratio is a comparison of two

Chapter 11.1 – Ratios and Rates
A ratio is a comparison of two numbers, a and b, by division. The numbers a and b are called terms of
the ratio. A ratio can be expressed in three different ways.
1. Word form: a to b
2. Ratio form: a : b
3. Fraction form: a/b
Ratios are often expressed in simplest form.
An equation showing that two ratios are equal is called a proportion. To check if two ratios are
proportionate use the Cross Product Rule.
In the following proportion
a:b =c:d
the middle terms b and c are called the means. The end terms a and d are called the extremes. To be
proportionate the product of the means must equal the product of the extremes. So a*d = b*c.
When there is a relation between variables, x and y, such that y = ax, a ≠ 0, y is proportional to x. This
means that there is a constant ratio between the corresponding values of x and y. In these cases “a” is
called the constant of proportionality.
The graph of a proportional relation always contains the origin and has a slope equal to the constant of
proportionality.
A ratio that compares quantities measured in different types of units is a rate. A unit rate is a rate in
which the second term or denominator is 1. Speed is a rate that compares distance and time. A unit
rate is 55 miles per hour. To find the unit rate express a given rate in simplest form.
A unit price is a unit rate for an item of purchase. Using the unit price can help you determine what
purchase is a better deal. Example: Which has a lower cost, 8 oz of shampoo for $2.98 or 12 oz of
shampoo for $5.26? You can find what the cost of 1 oz is for each price to determine the better deal.
You can rename (or convert) rates from one unit measure to another by using conversion factors. A
conversion factor is a rate equal of equal quantities that is used to multiply a quantity when converting
from one unit to another. The process of analyzing units to decide which conversion factor to use is
called dimensional analysis. Example: renaming miles per hour to feet per second.
Many verbal problems can be solved by renaming rates from one unit to another. When comparing
rates you need to make sure you are comparing similar units.
When performing conversions make sure you include the units in the ratios to make sure that the units
divide out.
Chapter 11.2 – Apply Proportion to Scale Models
A scale is a ratio between two sets of measurements, such as 2 m : 1 km. A scale drawing or scale
model uses a scale to represent an object smaller than (a reduction), or larger than (an enlargement)
the actual object. A map is an example of a scale drawing. With a map the scale is the ratio of the
distance on the map to the corresponding actual distance.
Blue prints and scale replicas are other types of scale drawings and scale models.
Examples: A map of California has a scale of two inches : 75 miles.
Sacramento is about 169 miles from Yosemite National Park. About how far apart are they on the map.
2 inches= x
75miles 169 miles
2 inches*169 miles = x*75 miles
338 inches*miles = 75x*miles
75 miles
75miles
4.5 inches = x
Chapter 11.3 Calculate Relative Error
Percent of error and relative error are measures of accuracy.
When scientists need to compare the results of two different measurements, the absolute
difference between the values is of very little use. The magnitude of error of being off by
10 cm depends on whether you are measuring the length of a piece of paper or the distance from New
Orleans to Houston. To express the magnitude of the error (or deviation) between two measurements
scientists invariably use percent error .
If you are comparing your value to an accepted/actual value, you first subtract the two values so that
the difference you get is a positive number. This is called taking the absolute value of the difference.
Then you divide this result (the difference) by the accepted value to get a fraction, and then multiply by
100% to get the percent error.
So,
% error = | your result - accepted value | x 100 %
accepted value
Notice that the error is a positive number if the experimental value is too high, and is a negative
number if the experimental value is too low.
Example: A student measures the volume of a 2.50 liter container to be 2.38 liters.
What is the percent error in the student's measurement?
Ans.
% error = (2.50 liters - 2.38 liters) x 100%
2.50 liters
= (.12 liters) x 100%
2.50 liters
= .048 x 100%
= 4.8% error
Relative error is the percent of error written as a decimal. To find relative error use the percent of
error formula but do not multiply the results by 100.
Percent of error can become aggravated or compounded by multiple measurements. Multiplication of
inaccurate measurements increases the error.
Chapter 11.4 – Apply Percent to Algebraic Problems
A percent is a ratio that compares a number to 100. Percentage = rate * base (p= rb)
Let's look at a problem requiring us to find the percent of a give number.
Matthew has a part-time job as a salesperson at a computer store. He is paid a 2% commission on his
total sales. One week, his total sales were $7000. How much was his commission.
A commission is an amount you are paid on top of your salary – usually tied to amount of sales you
make.
To find how much Matthew made in commission we can use two methods.
Method 1: Write and solve a percent proportion.
p = 2
7000
100
100p = 14000
100
100
p = 140
So Matthew's commission was $140.
Method 2: Use the percentage formula.
Percentage = rate * base
p = 2% * 7000
we never use percents in calculations we always convert to a decimal first.
P = .02 * 7000
p = 140
So, again we see that Aran's commission was $140.
If T = tax
MP = marked Price
SP = sales price, then
D = discount
T = r*MP (tax = rate* marked price)
TC = MP + T (total cost = marked price + tax)
D = r *LP (discount = rate * list price)
SP = LP – D (sales price = list price – discount)
TC = total cost
LP = list price
Interest is the amount of money charged (earned) for borrowing (saving or investing) money – which
is defined as principal. Simple interest is paid only on principal.
If P = principal
r = rate
t = time
I – interest
B = balance, then
I = Prt (interest = principal * rate * time)
B = P + I (balance = principal + interest)
The percent of change is the percent, or rate, a quantity increases or decreases from its original
amount.
Percent of Change = amount of change * 100
original amount
When you want to find the percent increase or decrease you use the percent of change formula. If the
new amount is greater than the original amount, it is a percent increase. If the new amount is less than
the original amount then you have a percent decrease.
Chapter 11.5 - The Trigonometric Ratios
The word trigonometry means the measure of triangles. The study of trigonometry involves the
relationship among the sides and angles of a triangle.
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The three basic
trigonometric ratios are sine, cosine and tangent. They are abbreviated as sin, cos and tan,
respectively.
Using Right Triangle
Sin A = length of the leg opposite <A = a
length of hypotenuse
c
Cos A = length of leg adjacent to <A = b
length of hypotenuse
c
Tan A = length of leg opposite to <A = a
length of leg adjacent to <A b
The acronym SOH CAH TOA is a way to remember the definitions of sine, cosine, and tangent.
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
You can use a calculator to find the values of sine, cosine and tangent ratios for a given angle. To do
this you plug in the angle measure and then push the sine, cosine or tangent button.
You can find the degree measure of an acute angle of a right triangle by using the inverse trig function
on a handheld calculator.
Chapter 11.6 Use Trigonometric Ratios to Solve Right Triangles
Given the length of two sides of a right triangle, you can use the trigonometric ratios to find the
measures of acute angles. You can use the trigonometric ratios or the Pythagorean Theorem to find the
length of the third side.
Solving right triangles
We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle,
that is, to find unknown parts in terms of known parts.




Pythagorean theorem: a2+ b2 = c2.
Sines: sin A = a/c, sin B = b/c.
Cosines: cos A = b/c, cos B = a/c.
Tangents: tan A= a/b, tan B = b/a.
Let's first look at some cases where we don't know all the sides. Suppose we don't know the hypotenuse
but we do know the other two sides. The Pythagorean theorem will give us the hypotenuse. For
instance, if a = 10 and b = 24, then c2 = a2 + b2 = 102 + 242 = 100 + 576 = 676. The square root of 676
is 26, so c = 26. (It's nice to give examples where the square roots come out whole numbers; in life they
usually don't.)
Now suppose we know the hypotenuse and one side, but have to find the other. For example, if b = 119
and c = 169, then a2 = c2 – b2 = 1692 – 1192 = 28561 – 14161 = 14400, and the square root of 14400 is
120, so a = 120.
We might only know one side but we also know an angle. For example, if the side a = 15 and the angle
A = 41°, we can use a sine and a tangent to find the hypotenuse and the other side. Since sin A = a/c, we
know c = a/sin A = 15/sin 41. Using a calculator, this is 15/0.6561 = 22.864. Also, tan A = a/b, so
b= a /tan A = 15/tan 41 = 15/0.8693 = 17.256. Whether you use a sine, cosine, or tangent depends on
which side and angle you know.
(from http://www.clarku.edu/~djoyce/trig/right.html)
Chapter 11.7 Use Trigonometric Ratios to Solve Verbal Problems
Two terms that are often used in problems about right triangles are angle of elevation and angle of
depression. Both angles are formed by an observers line of sight and a horizontal line.
Angles of Elevation and Depression
The angle of elevation of an object as seen by an observer is the angle between the horizontal and the
line from the object to the observer's eye (the line of sight).
If the object is below the level of the observer, then the angle between the horizontal and the observer's
line of sight is called the angle of depression.
Example: A building is 50 ft high. At a distance away from the building an observer notices that the
angle of elevation to the top of the building is 41º. How far is the observer from the base of the
building?
Step One:
Make a right triangle diagram. A drawing is shown below. The two legs of the right triangle are formed
by the building and the distance between the building and the observer.
Step Two:
Label the known distances in the diagram. The building is 50 feet high and the angle from
the observer to the top of the building is 41º. This is the angle of elevation.
Step Three:
Choose a variable to represent the unknown distance from the observer to the building. We
have chosen x to represent this distance.
Step Four:
Set up a trigonometric ratio involving sine, cosine, or tangent and the labeled
measurements. Notice that the building is across from or “opposite” the angle. Since the
building which is 50 feet and the distance from the observer which is x form the opposite
and adjacent sides of the triangle with respect to the 41º angle, we have .