Ratios that are equal are said to be proportionate to each other, such as 5/7 and 25/35.
A bowl contains a mixture of 1 cup cornstarch and 4 cups flour. A second bowl contains the same ingredients in the same proportion, but instead has 1.5 cups of cornstarch. If 12 cup of oats is added to the second bowl, what is the ratio of oats to flour in the second bowl?
First, determine the amount of flour in the second bowl, using the same ratio in the first bowl (which is 1 : 4). We represent the amount of flour with the variable x. Ratios are in proportion if they are equal, so:
1/4 = 1.5/x
x = 6
The ratio of oats to flour, then is 0.5 is to 6, or 1 : 12.
Questions involving ratio and proportion usually ask for an unknown part or component. See this example:
The concrete mix to a floor slab must follow the ratio of 1 part cement, to 3 parts sand, to 3 parts stone aggregates. How many buckets of cement (C) will be needed for a total of 10 buckets of sand and 10 buckets of stone aggregates?
The ratio is 1 : 3 : 3, and we need to find C for C : 10 : 10.
We may just use the first two parts of each ratio and set the two ratios equal to each other:
1/3 = C/10
10/3 = C
C = 3 and 1/3
Andy usually buys large Grade A eggs from her supplier at $2.50 per dozen. She chanced upon a farm where the eggs were sold at 70% of this price and bought 10 dozens. How much more would she have paid for the 10 dozens of eggs had she bought from her regular supplier?
First, calculate the price of eggs at the farm, which is 70% of $2.50:
Price at the farm = 0.7 * 2.50 = $1.75
Then, we proceed to answer the question being asked. We need to know the price she paid for the whole purchase, the price she would have paid had she bought from her regular supplier, and then solve for the amount she saved.
$1.75 is the price for one dozen. She bought ten dozens, so the amount she paid was:
Total amount paid = 1.75 * 10 = $17.50
The price for ten dozen eggs from her regular supplier:
Price from regular supplier = 2.50 * 10 = $25.00
Getting the difference of the two total amounts, we get Andy’s savings:
Amount Andy saved = 25 – 17.50 = $7.50
Familiarity with the unit conversion method, also called the factor-label method, comes in very handy when dealing with many rate questions and when double-checking answers. Let’s use this method in an actual rate question.
Alia has an annual basic salary rate of $53,760. If she works 8 hours a day, four days in a week, and four weeks in a month, what is her hourly rate?
Even without a formula, this can be solved using the unit conversion method.
Start with the given information and take note of the final unit of measure being asked in the question:
$53,760/year = __/hour
Using the unit conversion method, simply multiply the given with the conversion factors until you get to the required unit of measurement:
($53760)/(year) * (1 year)/(12 months) * (1 month)/(4 weeks) * (1 week)/(4 days) * (1 day)/(8 hours) = $35/hour
Take note that we deliberately wrote conversion factors in such a manner that similar units canceled out, for example, week units were written as numerator and denominator so that they canceled each other out. This is to say that you may write conversion factors in any way that makes it favorable for you to cancel units out.
The same is true with the other units we wanted to disappear, leaving us only with the units required in the answer. Also, the conversion factors used were those specifically given in the question, and not those we typically know, such as
1 day = 24 hours.
You need to sharpen your skills on matching graphs to the properties and values of a data set.
Categorical data, such as the genre of music preferred by students in a high school, are appropriately presented by pie and bar graphs.
Numerical data, which are either discrete or continuous, are plotted using line graphs, histograms, and scatterplots, such as Company A’s human resource department’s annual expenditures over a 10-year period.
On the SAT exam, you may be given a graph and asked to interpret it. You will need to understand what you see graphically and relate this to important features—the center, spread, and shape.
Inferences about the population can be made from the results of sample surveys as long as random sampling has been used for the study.
Take this statement for example:
In a survey based on a random sample of students in XY Senior High School, 68% said that they spend at least 7 hours a day on social networks.
You can correctly infer that “About 68% of all students in XY Senior High School spend at least 7 hours a day on social networks.”
Here’s another inference that can be made:
If the random sample consisted of 100 students out of 1,250 students and 12 of those surveyed said that they spent less than 3 hours a day on social networks, about how many students at XY Senior High spend less than 3 hours a day on social networks?
A reasonable estimate would be 150 students:
12/100 = N/1250
N = 150
A well-designed survey makes it possible to cut the cost and time necessary for a census and still obtain a result that can be generalized to the entire population. Through random sampling, all elements of the population have a probability of being selected, and the result is protected from biases.
Both experiments and observational studies investigate the causal relationship between a dependent variable and an independent variable. In experiments, the researcher has control over the participant’s assignment to groups and treatments given to each group; hence, random assignment of subjects can be done. This control is absent in observational studies, and findings derived from this method cannot be used for causal inference and generalization to the larger population.
It is important to know how the data in a study was obtained because this largely determines whether it is appropriate to draw and apply conclusions to the entire population.
Data are obtained by conducting a census (entire population), survey (random sample), experimental study (cause-and-effect, controlled), and observational study (cause-and-effect, not controlled).
Interest is the amount paid or earned for the use of money, such as in money borrowed or invested. Here’s an example:
Leila borrows $10,000 from a bank that charges at interest rate of 10% per year.
Here are the terms that we need to understand:
Principal — the amount borrowed or invested. In the example, the principal is $10,000.
Interest rate — the ratio of the amount paid for the use of money to the principal; it is customary to present this as percentage per year. In the example, the interest rate (also often called the annual interest rate) is 10%.
Time—the time or duration that the principal amount is borrowed or invested.
Interest—the amount paid for borrowing money or earned for investing money. Interest can be computed as simple interest or compound interest.
Simple interest—the amount paid for the use of money based on an annual interest rate. The formula for simple interest is:
Interest = principal * rate * time
where rate is the annual interest rate expressed in decimal form, and time is expressed in some measurement of time (normally years).
In the example for a loan duration of one year, the interest charged on the loan is:
Interest = 10,000 * 0.10 * 1 = $1,000.00
Take note that, if a loan’s duration is 2 years, the interest charged would be $2,000.
Questions involving compound interest basically entail these steps:
Solve for the interest in the first period.
Add the computed interest in the first period to the principal.
Use this total to solve for the interest in the second period.
Add the computed value to the previous total.
Repeat the procedure as many times as the number of years desired (such as 5 times for the interest in 5 years, compounded annually).
Add all interest for the different periods to get the total interest of the amount borrowed compounded annually.
Using the same example as above, the interest for the first year will be the same at $1,000.
For the second year, it will be:
Interest = (10,000 + 1,000) * 0.10 * (1) = 1,100
Adding all interests for 2 years, the total interest charged would be $2,100. Take note that this is a bigger amount than the simple interest on the same principle for the same period of time.
However, when the period of time (loan duration or term or investment term) involves bigger numbers, it is more practical to use this formula:
A = P * (1+r)t
where A is the future value to be paid, P is the principal, r is the annual interest rate in decimal form, and t is time or period in years.
When a question states ”10% monthly compounding”, it does not mean 10% per month. Rather, it means, a monthly interest of 0.83% (10% divided by 12), compounded monthly.
The formula varies a bit:
A = P * (1 + r/n)nt
where n is the number of times the interest compounds each year.
In the given example of monthly compounding interest and using the same principle ($10,000), the interest for 2 years is $2,107.00.
A = 10,000 * (1+0.10/12) 12 * 2 = 12,107
Interest = 12,107 – 10,000 = 2,107
Data analysis is more applicable to real life than many people think. When an ad for a pet lice shampoo says that “This shampoo is recommended by 9 out of 10 vets”, what does it mean? Does it mean that 90% of all vets recommend that particular shampoo for their pets? Or is that a valid claim at all? How was that statement arrived at?
A public opinion survey says that 65% of people support a tax increase. What was the age group surveyed? How representative was it of the whole population?
In a study that says that the leading cause of low productivity in a workplace is absenteeism, see first how the data was collected. It is highly likely that this was done using an observational study. This method of data collection is very useful for understanding how one variable affects another. Because the method lacks the feature of randomness, however, it will be inappropriate to generalize the results to the larger population.
In real life, you get a better grasp of data and how they are being applied by taking them with a grain of salt and asking how those data were produced, collected, presented, and analyzed.
Probability is the likelihood of an event occurring. The value is given as a fraction or a decimal number, and it is always less than 1. The closer a probability is to 1, the higher its probability of occurring. It is computed with this formula:
Probability = number of desired outcomes/total number of possible outcomes
To determine the probability of randomly picking a green shirt from a hamper that contains 2 green shirts, 3 red shirts, 5 yellow shirts, and 5 blue shirts, we apply the formula and calculate probability:
P(green shirt) = 2/15
There is a 2/15 or 0.133 chance of picking up a green shirt.
To determine the probability of two or more events, we multiply their individual probabilities. The probability of randomly picking a green shirt and then a red shirt from the same example is calculated as follows:
Probability = 2/15 * 3/14 = 6/210 = 1/35
Note that when calculating the total number of shirts in the hamper for determining the probability of choosing a red shirt, the total number is 14 because a green shirt was already selected and removed from the hamper.