SAT Math Lesson-03

SAT Math Lesson-03

Word Problems

(Ratios & Proportions)

Simple concepts & scenarios:

There are two types of ratios namely: 1). Simple ratios, 2). Compound ratios
 

1). Simple ratios:

Simple ratios involves two variables. For instance, let’s suppose Boys to Girls ratio in a school is 4:3. There are two ways to write this ratio as bellow:

Ad. Q2-1

Important Note: Ratios doesn’t give numbers. It just tells the proportion of a variable (e.g boys) with respect to the other variable (e.g. girls).

For instance, in example above, if there are 4 boys, then there would be 3 girls and vice versa. If there are (4 × 2 = 8) boys, then there would be (3 × 2 = 6) girls in same proportion, and so on.

In short, from the given ratio, we can say that number of boys are multiple of 4, while number of girls are the (same integer) multiple of 3. For instance, if there are 4 × 5 = 20 boys, then there would be 3 × 5 = 15 girls. So the integer which is multiplying by 4 must also be the same integer which multiply with 3. But we are not certain from the given ratio that how many boys or how many girls are there.

So we concluded that ratios doesn’t tell numbers, it just give proportion.

Similarly the total number of students in the school cannot be find from the ratio of boys to girls. But total number of students in the school may be find as multiple of some integer. For instance in above example,

Boys   :  Girls

    4     :    3

As number of boys are multiple of 4 and number of girls are the same multiple of 3, so let’s suppose that integer which multiply is x

⇒ 4x   :   3x

In this way we did converted ratio (i.e proportion) into numbers. i.e. there are 4x number of boys (not 4 boys); and there are 3x number of girls (not 3 girls).

Also,

⇒ 4x + 3x = 7x

So total number of students in the school must be multiple of 7.

Important Note: Always remember that in all ratios question, you must convert ratios into number by placing a variable with each ratio value, as given in last equation above.

 

Now, let’s learn how to find numbers from ratios after some changes in data given in question. For instance, in above example of boys and girls, If 12 boys removed from the school and 15 girls added, the new ratio of boys to girls become 1 : 1(i.e. number of boys & girls would become equal), then how many boys are present in the school?

Solution:

According to the given condition,

Boys   :  Girls

    4     :    3

4x   +   3x = 7x

Important Note: This above procedure is always required in almost every ratios questions.

Now, as it’s given that 12 boys removed, so

After 12 boys removed, number of boys would be 4x – 12. And after 15 girls addition, number of girls would become 3x + 15. Then it said that the new ratio become 1 : 1, so after change,

(4x – 12)(3x + 15) = 11

Now, by solving it for value of x, we’ll get

⇒ 4x – 12 = 3x + 15

x = 27

Now, we need to find number of boys in the school (at present), so we need to put value of x in 4x – 12 (present number of boys after changes)

⇒ Present number of boys = 4 (27) – 12

⇒ Present number of boys = 108 – 12

⇒ Present number of boys = 96 Answer

 

2). Compound ratios:

Compound ratio involves more than two variables. For instance, let’s suppose Boys to Girls ratio in a school is 4 : 3, and Staff to Girls ratio in that school is 2 : 9. Let’s learn how to find compound ratio of these three variable.

For that purpose, first we must look for the variable which comes twice in the given information of simple ratio form (i.e. in the question). Here Girls are that required variable. Now always place this variable at middle of the other two as bellow:

Boys : Girls : Staff

Now, as Boys : Girls = 4 : 3

and Girls : Staff = 9 : 2 (As Staff : Girls = 2 : 9, we need Girls : Staff that would be reversed)

Now, place the simple ratios in combined as follow:

Ad. Q2-2

 
 
 
 
 
 

After that always use this way to make compound ratio as follows:

Ad. Q2-3

 
 
 
 
 
 
 
 

We have done the above step in order to have same numbers in ‘number of girls’ on both separate ratios. So whole this calculation is required to do so.

By taking 3 common from the above ratios,

⇒ Boys   :   Girls   :   Staff     =     12   :   9   :   2                 (By taking 3 common from all three ratios)

So finally we got compound ratio as follows:

⇒ Boys   :   Girls   :   Staff
      12     :     9     :     2

Again, As I said, always do convert the ratios into numbers by multiplying with a variable, i.e.

⇒ Boys   :   Girls   :   Staff
      12x   :     9x     :     2x

⇒   12x   +   9x   +   2x   =   23x

So there are total 23x people in the school.

If in question, the two split simple ratios is given in addition to total number of people in the school is 46, then it was asked to find the number of boys in the school, how to find? Let’s learn this.

At first, you need to convert that two simple ratios into compound ratio as shown above. Then convert it into number with variable multiple (e.g x as we did above). So at this stage you are exactly on the point that we’ve done before i.e

⇒ Boys   :   Girls   :   Staff

      12x   :     9x     :     2x

⇒   12x   +   9x   +   2x   =   23x

Now, as total people in the school = 23x = 46

x = 2

As number of boys are 12x that we need to find, so by putting value of x here

⇒ Number of boys = 12 (2)

⇒ Number of boys = 24 Answer

 

Similarly,

Now in addition to the two simple ratios, if it is said that, for instance, the number of boys in the school reduce by 90, while staff would increase by 20, making the ratio of Boys to Staff half as much as it was before. Then how many Girls are in the school?

To answer this, as I told you, we first need to solve for the compound ratios of the three variable as we did before and we’ll always do this step in compound ratios. After doing this we’ll get,

⇒ Boys   :   Girls   :   Staff

      12x   :     9x     :     2x —————- (eq. 1)

⇒   12x   +   9x   +   2x   =   23x

Now, as it’s said number of Boys reduced by 90 and Staff increased by 20. And we see that Boys were 12x, while Staff were 2x before the changes mentioned in the question.

After changes,

⇒ Boys = 12x – 90       &       Staff = 2x + 20

Now, after the changes, it is said that the ratio of Boys to Staff would become half as much as it was before the changes, so

Ad. Q2-4

 
 
 
 
 
 
 
 
 
 
 
 
 

⇒ 2(12x – 90) = 12(x + 10)

⇒ 12x – 90 = 6(x + 10)

⇒ 6x = 150

x = 25

Now as we need to find the total number of Girls in the school, which are according to (eq. 1) 9x

⇒ Number of Girls = 9x

By putting value of x, we’ll get,

⇒ Number of Girls = 9(25)

⇒ Number of Girls = 225 Answer

 

Advance level concepts & scenarios:

Now let’s discuss four variable scenarios. This scenario only came in GMAT exam till date, but this may come in GRE as well in future. So let’s discuss this.

 

If the ratio of grade A students to grade B students is 3 : 4 which is twice as much as the ratio of grade B students to grade C. If grade A students to grade D students are in ratio of 5 : 9, what is the ratio of grade C students and grade D students?

 

Solution:

First we need to find A : B : C by same method as discussed earlier, while doing so, we’ll finally get

G Q2-5

 
 
 
 
 
 

Now, we have to adjust this in such a way that A must come between D and B, because A came twice while introducing the fourth variable. So now we need to find D : A : B, while doing so, we’ll get,

G Q2-6

 
 
 
 
 
 

At this stage, we need to make A : B same numbers on both of the compound ratios in A : B : C and in D : A : B.

For that purpose, let’s divide the A : B : C by 3, and then multiply by 5. We did this because we have to make A : B to be 15 : 20, the same as the ratio of A to B in D : A : B in above.

Therefore, after dividing A : B : C by 3 we’ll get:

A : B : C
3 : 4 : (323)

Now, by multiplying this by 5, we’ll get

A : B : C
15 : 20 : (1603)

All in all:

A  :   B :  C  :   D
15 : 20 : (1603) :27

By multiplying whole ratio by 3, to eliminate fraction,

A  :   B :  C  :   D
45 : 60 : 160 : 81

Therefore,

A   : D
45 : 81

By dividing by 9,

A : D
5 : 9 Answer

 

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