SAT MATH: PROBLEM SOLVING AND DATA ANALYSIS
Problem solving and data analysis focuses on real world problems like ratios, data handling and unit conversion.
# RATIOS PROPORTION AND PERCENTAGE
RATIOS
Well this section is too easy to explain but still i will do as it's my responsibility.
Starting from an example, let there be a class in which the ratio of girls is to boy is 2:3 and if total number of students is 50, then find number of girls.
ANS) number of girls/number of boys = 2/3
rearranging them
no. of girls/2 = no. of boys/3
let it be equal to X
no. of girls/2 = no. of boys/3 = X
So, number of girls = 2X; number of boys = 3X
total number of students = 2X + 3X = 5X = 50
this implies X = 10
no. of girls = 2X = 20
PROPORTIONS
There are 2 types of proportions Direct proportion and Indirect proportion.
1)DIRECT PROPORTION
In direct proportion when 1 increases the other one increases too.
Ex - the more number of people will create a product the more product will be produced.
MAN POWER ∝ PRODUCTION
EX) If 10 people can create 70 bulbs a day, then how many bulbs will 30 people make in 1 day.
Ans) Through common sense we can understand that as number of people will increase the production will increase.
10 people create 70 bulbs a day
1 person creates 7 bulbs a day
so, 30 people will create 7 ✕ 30 bulbs a day = 210 bulbs a day
2) INVERSE PROPORTION
In inverse proportion when 1 increases the other decreases.
EX - The more number of people will work on a target the lesser time it will take.
MAN POWER ∝ 1/TIME TAKEN
Notice in this example we will have to reduce time not increase the production.
EX) If it takes 2 people 6 days to paint the boundaries of a stadium then how many days will it take 3 people to do the same.
ANS) MAN POWER ∝ 1/TIME TAKEN
2 people do it in 6 days
1 person will do it in 12 days
3 people will do it in 12/3 days = 4
So, it will take 3 people 4 days to paint the boundaries.
#PERCENTAGE
Mostly comprises of profit and loss questions. So, let's understand the concept from one of these questions.
EX) You bought a table of $100 at a 20% discount and then sold it to someone making profit of 30% on your cost price. Find the selling price
ANS) Since the table is of 100% so at 20% discount you will get it of 100(1 - 20/100) = 100(0.8) = $80
Now you sold it for a profit of 30% on your cost price
Profit = Selling price - Cost price
30% of cost price = Selling price - Cost price
cost price ✕ 130% = selling price
selling price = 80 ✕ 130/100 = $104
So, selling price = $104
Notice % is called as per cent referring per century thus every time you remove this sign you will have to divide by 100 and if you want to insert it then multiply by 100.
Here, Profit = Selling price - cost price
We got this value as positive thus it was considered as profit if it was negative then it would have been considered as loss.
So, Profit or Loss = Selling price - Cost price
Whether it's profit or loss depends on the value.
UNIT CONVERSION
Lucky for you in unit conversion the conversion values are given in question thus you won't have to remember any.
The most common one is between miles and KM
EX) A road is of 0.8 miles long then find how long will it be in KMs.(1 mile = 1.6 KM)
ANS) I took a very basic example as there is no point to take a tough one as all we have to is to convert the unit.
1 mile = 1.6 KM
Dividing both sides by 1.6
1 KM = 1/1.6 mile
Multiplying by 8 on both sides
8 kMs = 8/1.6 mile = 5 miles
In exam, I want you to do it in this way,
1 mile = 1.6 KM → 1 KM = 1/1.6 mile
this part (you have to do this part in your mind)
now to convert 8 KMs into mile
8 KMs = 8/1.6 mile = 5 mile
Be real quick with these questions. Just have a basic common sense that since mile is bigger than KM then the answer would be less than 8. It may happen that 5 be the only option lesser than 8. In that case, you could eliminate all others.
#DENSITY
Here we are not talking about Mass/Volume, what we are discussing about are other densities like population density.
EX) In a city A, resides a population of 10,000 in an area of 20 sq. miles. While in city B, the population is 20,000 and the area of 10 sq. miles.
Here population density of city A
= population of city A/area of city a
= 10,000 people/20 sq miles
= 500 people per sq miles
population density of city B
= 20,000 people/ 10 sq. miles =2000 people per square miles
So to take the population of both the city we can't take the average. can you tell me why?
It's because average is taken of linear functions only
So to calculate the population density we have to divide total population by total area
You may also be given the population density of city A and city B and then asked the population density of both the city combined. In that case you should first find the area and population of both the cities and solve as shown above.
PROBABILITY
Here i will try to refrain myself from discussing cards and dices in probability since you must be already bored with them.
So let's begin with Probability
So let's take an example there are 20 balls in a bag 8 are green and 12 are red. What is the probability of taking out a green ball.
Ans) Probability = favorable outcome/total outcome = 8/20 = 0.4
I have already told you that we can convert anything in percentage form by multiplying it by 100
So probability = 0.4 × 100% = 40%
Here it depends on the options as to which form is the correct answer.
There can be questions where the data be written in a table and you would be scared at the sight of table but just remember you have to find the total outcome and favorable outcome. There may be a condition too in that case find total outcome of that condition only.
EX) In a hotel, the general manager found an employee to be late and looking at the data of yesterday he asks you to find the probability of an employee to be of housekeeping dept if he is late.
Since he is late so total outcome = 15
Now the person being housekeeping = 9
so probability = 9/15 = 0.6
There can be another type of questions like probability of a ball falling in a specific part of park.
In that case your Favorable outcome outcome/total outcome
becomes Favorable area/Total area
STATISTICS
It's used to analyze the given data. Let's start with mean.
MEAN - It's the sum of all the values divided by total number of values.
Kind of finding the average
MEDIAN - When you arrange all the data in ordered form then the middle number is the mean
MODE - The most frequently occurring number is known as mode
RANGE - Difference between the maximum and minimum value
OUTLIER - It's a value that significantly differ from other values causing inconsistency in statistical analysis.
# Time to explain above points to you through example. let there be an exam of 100 marks given by 11 students and it's values be given below.
1) Let's calculate the mean
Mean = Sum of all the values/total number of values
Number of values = N = 11
sum = S = 85 + 76 + 59 + 85 + 92 + 84 + 91 + 78 + 63 + 72 + 85
= 870
mean = S/N = 870/11 = Approx 79
2) MEDIAN
Notice that in the definition of median it's written that data should be arranged in the order.
So let's first arrange them in increasing order.
The median is the middle one for 11 numbers the middle one is going to be 6th term, which is 84 so median is 84.
For even number of terms median is N/2th and (N/2 + 1)th terms. Like if there were 12 terms then median would had been N/2th = 6th term and (N/2 + 1)th = 7th term
So, it would have 2 medians.
3) MODE - It is the most frequently occurring number. Here you can see 85 was occurring most frequently thus it is the mode. There may be more than 1 mode if those many values were appearing more frequently than others. Like if there were 3 times 84 in the above data then mode would had been 84 and 85. And if among the 3 values of 85, 2 were 86 and 87 then there would have been no mode in this data as no number was appearing more frequently.
4) RANGE - It's the maximum value - minimum value
= 92 - 59
= 33
Thus, the range is 33
# Now you could have also got it in tabular or graphical form. Let's discuss the tabular form as you can interpret the graph form in it. Most probably it will be a histogram.
a) mean = 14 median = 24
c) mean = 21 median = 32
d) mean = 8 median = 16
ANS) Here the data is already arranged in order to work out the median. Total number of students = 15+10+18+4 = 47
Median = ((N+1)/2)th term for odd
First 15 terms are between 0 - 10 then next 10 terms are between 10 - 20 thus 24th term is between 10 - 20
So the median is between 10 - 20
Coming to the mean considering the minimum value it can attain -
In it the first 15 students attend 0 classes ( Min is 0 - 10 Range) then 10 students attend 10 classes, 18 attend 20 classes and 4 attend 30 classes it becomes
mean = (0 × 15 + 10 × 10 + 20 × 18 + 4 × 30)/47
= (100 + 360 + 120)/47 = 580/47 = 12.3
Hence mean is bigger than 12.3. Here we multiplied 0 by 15, 10 by 10 and so on because 0 is the value and 15 is the number of times that value came. We actually added 0 for 15 times as there were 15 students who we assumed to attend that many classes.
Hence the answer is B (only B satisfies all the conditions)
Similarly you can calculate maximum value by using right end of the Range like 10,20,30 and 40 multiplied by 15,10,18 and 4 respectively and divide by 47.
#NOTE - The Right end values belong to the next interval but we use it to calculate the approximate maximum value.
OUTLIERS
As discussed above they are the ones who don't follow trends of other values. they have a significantly different value. For ex in values - 10,13,12,17,85,21 and 14 the number 85 is an outlier because of being too big they affect mean a lot.
STANDARD DEVIATION
It is a measure of amount of variation of values from the mean. a low standard deviation means the values are close to mean while a high standard deviation means that the values are spread out over a wide range.
Let's suppose there are 2 sets of 10 values In set A the mean is 26 and values are between 20 and 30 while in set B the mean is again 25 but the values range from 4 to 58 with only 3 values in 20 - 30 range. Thus we can say that value is more deviated or farther away in set B when compared to set A so Standard Deviation of set B will be greater than set A.
MARGIN OF ERROR
It's a statistic expressing the amount of random sampling error in data. The larger the Margin of Error the lesser will be the confidence in interpretation of data.
#KEY POINTS
1) If standard deviation increases then margin of error increases and vice versa. They are in direct proportion.
2) If sample size ( number of values) increases then margin of error decreases and vice versa.
3) Margin of error gives information about the mean not about individual data
In SAT you are expected to have just a basic understanding of Margin of Error and that is that if mean of sample is 400 and margin of error is 20 then it means that -
It is plausible that mean is between 380(44 - 20) and 420(400 + 20)
That's all you need to know about it there is no need to go in depth.
SAT HOMEPAGE
SAT EXAM FORMAT
SAT READING
SAT WRITING
SAT MATH
SAT MATH: HEART OF ALGEBRA
SAT MATH: PROBLEM SOLVING AND DATA ANALYSIS
SAT MATH: ADDITIONAL TOPIC IN MATH
SAT MATH: PASSPORT TO ADVANCE MATH
SAT MATH:GRAPH
SAT ESSAY
# RATIOS PROPORTION AND PERCENTAGE
RATIOS
Well this section is too easy to explain but still i will do as it's my responsibility.
Starting from an example, let there be a class in which the ratio of girls is to boy is 2:3 and if total number of students is 50, then find number of girls.
ANS) number of girls/number of boys = 2/3
rearranging them
no. of girls/2 = no. of boys/3
let it be equal to X
no. of girls/2 = no. of boys/3 = X
So, number of girls = 2X; number of boys = 3X
total number of students = 2X + 3X = 5X = 50
this implies X = 10
no. of girls = 2X = 20
PROPORTIONS
There are 2 types of proportions Direct proportion and Indirect proportion.
1)DIRECT PROPORTION
In direct proportion when 1 increases the other one increases too.
Ex - the more number of people will create a product the more product will be produced.
MAN POWER ∝ PRODUCTION
EX) If 10 people can create 70 bulbs a day, then how many bulbs will 30 people make in 1 day.
Ans) Through common sense we can understand that as number of people will increase the production will increase.
10 people create 70 bulbs a day
1 person creates 7 bulbs a day
so, 30 people will create 7 ✕ 30 bulbs a day = 210 bulbs a day
2) INVERSE PROPORTION
In inverse proportion when 1 increases the other decreases.
EX - The more number of people will work on a target the lesser time it will take.
MAN POWER ∝ 1/TIME TAKEN
Notice in this example we will have to reduce time not increase the production.
EX) If it takes 2 people 6 days to paint the boundaries of a stadium then how many days will it take 3 people to do the same.
ANS) MAN POWER ∝ 1/TIME TAKEN
2 people do it in 6 days
1 person will do it in 12 days
3 people will do it in 12/3 days = 4
So, it will take 3 people 4 days to paint the boundaries.
#PERCENTAGE
Mostly comprises of profit and loss questions. So, let's understand the concept from one of these questions.
EX) You bought a table of $100 at a 20% discount and then sold it to someone making profit of 30% on your cost price. Find the selling price
ANS) Since the table is of 100% so at 20% discount you will get it of 100(1 - 20/100) = 100(0.8) = $80
Now you sold it for a profit of 30% on your cost price
Profit = Selling price - Cost price
30% of cost price = Selling price - Cost price
cost price ✕ 130% = selling price
selling price = 80 ✕ 130/100 = $104
So, selling price = $104
Notice % is called as per cent referring per century thus every time you remove this sign you will have to divide by 100 and if you want to insert it then multiply by 100.
Here, Profit = Selling price - cost price
We got this value as positive thus it was considered as profit if it was negative then it would have been considered as loss.
So, Profit or Loss = Selling price - Cost price
Whether it's profit or loss depends on the value.
UNIT CONVERSION
Lucky for you in unit conversion the conversion values are given in question thus you won't have to remember any.
The most common one is between miles and KM
EX) A road is of 0.8 miles long then find how long will it be in KMs.(1 mile = 1.6 KM)
ANS) I took a very basic example as there is no point to take a tough one as all we have to is to convert the unit.
1 mile = 1.6 KM
Dividing both sides by 1.6
1 KM = 1/1.6 mile
Multiplying by 8 on both sides
8 kMs = 8/1.6 mile = 5 miles
In exam, I want you to do it in this way,
1 mile = 1.6 KM → 1 KM = 1/1.6 mile
this part (you have to do this part in your mind)
now to convert 8 KMs into mile
8 KMs = 8/1.6 mile = 5 mile
Be real quick with these questions. Just have a basic common sense that since mile is bigger than KM then the answer would be less than 8. It may happen that 5 be the only option lesser than 8. In that case, you could eliminate all others.
#DENSITY
Here we are not talking about Mass/Volume, what we are discussing about are other densities like population density.
EX) In a city A, resides a population of 10,000 in an area of 20 sq. miles. While in city B, the population is 20,000 and the area of 10 sq. miles.
Here population density of city A
= population of city A/area of city a
= 10,000 people/20 sq miles
= 500 people per sq miles
population density of city B
= 20,000 people/ 10 sq. miles =2000 people per square miles
So to take the population of both the city we can't take the average. can you tell me why?
It's because average is taken of linear functions only
So to calculate the population density we have to divide total population by total area
PROBABILITY
Here i will try to refrain myself from discussing cards and dices in probability since you must be already bored with them.
So let's begin with Probability
So let's take an example there are 20 balls in a bag 8 are green and 12 are red. What is the probability of taking out a green ball.
Ans) Probability = favorable outcome/total outcome = 8/20 = 0.4
I have already told you that we can convert anything in percentage form by multiplying it by 100
So probability = 0.4 × 100% = 40%
Here it depends on the options as to which form is the correct answer.
There can be questions where the data be written in a table and you would be scared at the sight of table but just remember you have to find the total outcome and favorable outcome. There may be a condition too in that case find total outcome of that condition only.
EX) In a hotel, the general manager found an employee to be late and looking at the data of yesterday he asks you to find the probability of an employee to be of housekeeping dept if he is late.
Since he is late so total outcome = 15
Now the person being housekeeping = 9
so probability = 9/15 = 0.6
There can be another type of questions like probability of a ball falling in a specific part of park.
In that case your Favorable outcome outcome/total outcome
becomes Favorable area/Total area
STATISTICS
It's used to analyze the given data. Let's start with mean.
MEAN - It's the sum of all the values divided by total number of values.
Kind of finding the average
MEDIAN - When you arrange all the data in ordered form then the middle number is the mean
MODE - The most frequently occurring number is known as mode
RANGE - Difference between the maximum and minimum value
OUTLIER - It's a value that significantly differ from other values causing inconsistency in statistical analysis.
# Time to explain above points to you through example. let there be an exam of 100 marks given by 11 students and it's values be given below.
Mean = Sum of all the values/total number of values
Number of values = N = 11
sum = S = 85 + 76 + 59 + 85 + 92 + 84 + 91 + 78 + 63 + 72 + 85
= 870
mean = S/N = 870/11 = Approx 79
2) MEDIAN
Notice that in the definition of median it's written that data should be arranged in the order.
So let's first arrange them in increasing order.
The median is the middle one for 11 numbers the middle one is going to be 6th term, which is 84 so median is 84.
For even number of terms median is N/2th and (N/2 + 1)th terms. Like if there were 12 terms then median would had been N/2th = 6th term and (N/2 + 1)th = 7th term
So, it would have 2 medians.
3) MODE - It is the most frequently occurring number. Here you can see 85 was occurring most frequently thus it is the mode. There may be more than 1 mode if those many values were appearing more frequently than others. Like if there were 3 times 84 in the above data then mode would had been 84 and 85. And if among the 3 values of 85, 2 were 86 and 87 then there would have been no mode in this data as no number was appearing more frequently.
4) RANGE - It's the maximum value - minimum value
= 92 - 59
= 33
Thus, the range is 33
# Now you could have also got it in tabular or graphical form. Let's discuss the tabular form as you can interpret the graph form in it. Most probably it will be a histogram.
c) mean = 21 median = 32
d) mean = 8 median = 16
ANS) Here the data is already arranged in order to work out the median. Total number of students = 15+10+18+4 = 47
Median = ((N+1)/2)th term for odd
= ((47+1)/2)th
=
24th term
First 15 terms are between 0 - 10 then next 10 terms are between 10 - 20 thus 24th term is between 10 - 20
So the median is between 10 - 20
Coming to the mean considering the minimum value it can attain -
In it the first 15 students attend 0 classes ( Min is 0 - 10 Range) then 10 students attend 10 classes, 18 attend 20 classes and 4 attend 30 classes it becomes
mean = (0 × 15 + 10 × 10 + 20 × 18 + 4 × 30)/47
= (100 + 360 + 120)/47 = 580/47 = 12.3
Hence mean is bigger than 12.3. Here we multiplied 0 by 15, 10 by 10 and so on because 0 is the value and 15 is the number of times that value came. We actually added 0 for 15 times as there were 15 students who we assumed to attend that many classes.
Hence the answer is B (only B satisfies all the conditions)
Similarly you can calculate maximum value by using right end of the Range like 10,20,30 and 40 multiplied by 15,10,18 and 4 respectively and divide by 47.
#NOTE - The Right end values belong to the next interval but we use it to calculate the approximate maximum value.
OUTLIERS
As discussed above they are the ones who don't follow trends of other values. they have a significantly different value. For ex in values - 10,13,12,17,85,21 and 14 the number 85 is an outlier because of being too big they affect mean a lot.
STANDARD DEVIATION
It is a measure of amount of variation of values from the mean. a low standard deviation means the values are close to mean while a high standard deviation means that the values are spread out over a wide range.
Let's suppose there are 2 sets of 10 values In set A the mean is 26 and values are between 20 and 30 while in set B the mean is again 25 but the values range from 4 to 58 with only 3 values in 20 - 30 range. Thus we can say that value is more deviated or farther away in set B when compared to set A so Standard Deviation of set B will be greater than set A.
MARGIN OF ERROR
It's a statistic expressing the amount of random sampling error in data. The larger the Margin of Error the lesser will be the confidence in interpretation of data.
#KEY POINTS
1) If standard deviation increases then margin of error increases and vice versa. They are in direct proportion.
2) If sample size ( number of values) increases then margin of error decreases and vice versa.
3) Margin of error gives information about the mean not about individual data
In SAT you are expected to have just a basic understanding of Margin of Error and that is that if mean of sample is 400 and margin of error is 20 then it means that -
It is plausible that mean is between 380(44 - 20) and 420(400 + 20)
That's all you need to know about it there is no need to go in depth.
BONNE CHANCE !
SAT HOMEPAGE
SAT EXAM FORMAT
SAT READING
SAT WRITING
SAT MATH
SAT MATH: HEART OF ALGEBRA
SAT MATH: PROBLEM SOLVING AND DATA ANALYSIS
SAT MATH: ADDITIONAL TOPIC IN MATH
SAT MATH: PASSPORT TO ADVANCE MATH
SAT MATH:GRAPH
SAT ESSAY
Comments