MCQs about ratio:

1:

If the ratio of boys to girls at a school picnic is 5:3, which of the following cannot be the number of children at the picnic?

A.  24

B.  40

C.   96

D.  150

E.   720

2:

If 2/3 of the workers in an office are nonsmokers, what is the ratio of smokers to nonsmokers?

A.  2:5

B.  1:2

C.   3:5

D.  2:3

E.   3:2

3:

What is the ratio of circumference of a circle to its radius?

A.  1

B.  π/2

C.   (π) ^½

D.   π

E.   2 π

Answer1:

If 5x and 3x are the number of boys and the number of girls, respectively, at the picnic, then the number of children present is 5x+3x=8x. Therefore, the number of children must be a multiple of 8. Only 150 is not divisible by 8.

Answer2:

Of every 3 workers, 2 are nonsmokers, and 1 is a smoker. Then, the ratio of smokers to non-smokers is 1:2

Answer3:

As π is the ratio of circumference to the diameter of a circle, so

π=  c/d=  c/2r

2π= c/r

E

Questions about Arithmetic

A 1:

If the sum of 3 consecutive integers is less than 75, what is the greatest possible value of the smallest of the three integers?

A 2:

If 2<x<4 and 3<y<7, what is the largest integer value of x+y?

A 3:

What is the smallest number that is divisible by both 34 and 35?

Answer  1:

Let the numbers be: n, n+1 , n+2. Then

n+ (n+1)+(n+2)+(n+3)=3n+3

Now by the given condition:

3n+3<75⇒3n<72⇒n<24

The most n can be is 23.

Now try three consecutive numbers whose sum is less than 75.

So, the numbers must be 23,24,25

Answer is 23.

Answer 2:

If x and y are integers, the largest value is 3+6=9. However, although x+y is to be an integer, neither x nor y must be. If x=3.8 and y=6.2, then x+y=10

So answer=10

Answer 3:

You are being asked for the LCM (least common multiple) of 34 and 35.

So the LCM of the numbers is 34×35=1190

MCQs about percents:

1:

What is 10% of 20% of 30%?

A.    0.006%

B.    0.6%

C.     6%

D.    60%

E.     6000%

2:

If 1 micron=10,000 angstroms, then 100 angstroms is what percent of 10 microns?

A.    0.0001%

B.    0.001%

C.     0.01%

D.    0.1%

E.     1%

Answer1:

10% of 20% of 30% = .10 ×.20×.30=0.006=0.6 %  (B)

Answer2:

1 micron=10,000 angstroms

10 micron=100,000 angstroms,  then,

Dividing both sides by 1000 gives

100 angstroms=  1/1000(10 microns ) and

                        1/1000=0.001=0.1%

                                                                                                                   

Questions about fractions:

B 1:

If 4/7 of the 350 sophomores at Punjab College are girls, and 7/8 of the girls play on a team, how many sophomore girls do not play on a team?

B 2:

In the meat department of a supermarket, 100 pounds of chopped meat was divided into packages, each of which weighed 4/7 pound. How many packages were there?

B 3:

In a jar, half of the marbles are red, ¼ are white, and 1/5 are blue. What fraction of the marbles are neither red, white, nor blue?

Answer1:

There are 4/7 ×350 =200 sophomore girls.

Of those, 7/8 ×200 play on a team. Then, 200-175=25 does not play on a team.

Answer2:

100÷  4/7= 100 ×   7/4=175

Answer3:

The red, white and blue marbles constitute

(1/2)+(1/4)+(1/5)=19/20

Of the total, so

1-(19/20)=1/20

1/20 of the marbles are neither red, white, nor blue.

Basic arithmetic concepts

A 1:

If A is the solution set of the equation x²-4=0 and B is the solution set of the equation x²-3x+2=0, how many elements are in the union of the two sets?

A 2:

What is the sum of the product and quotient of 7 and 7 ?

A 3:

What is the product of all the integers from -3 to 6 ?

Answer1:

 Solving each equation, you get A={-2,2} and B={1,2}.

 Therefore AUB={-2,1,2}. There are three elements in the union.

Answer2:

Product: 7×7=49. Quotient: 7÷7=1. Sum=49÷1=50.

So the answer is 50.

Answer3:

Before reaching to your calculator, think. You are asked for the product of 10 numbers, one of which is 0. Then the product is 0.

Important points Chapter 2 book 1

Logical statements are also called propositions

Negation is unairy operation.

Conditional is equivalent to contrapositive.

Converse is equivalent to inverse.

Tautology means ⇾always true.

Absurdity ⇾ always false.

Contingency ⇾either true or false.

MCQ:

(p⇾q) ⇾(∼q⇾∼p) is:

a)    Tautology

b)   Absurdity

c)    Contingency

d)   None of these

Don’t draw the table to check it. You are not provided much time to draw a table for this type of MCQs. First check the conditions carefully.

As conditional is equivalent to contrapositive so it is a tautology. (a)

Residue classes:

In residue class 4:

1)   0 is identity.

2)   3 is the inverse of 1.

3)   2 is the inverse of 2.

4)   1 is the inverse of 3.

Same is in the case of other residue classes depending upon the class.

Questions about ratio

1:

Frannie’s  Frozen Yogurt sells three flavors: vanilla, chocolate and coffee. One day, Frannie sold 240 cones, and the ratio of vanilla to chocolate to coffee was 8:17:15. How many chocolate cones were sold that day?

Answer:

Let 8x,17x, and 15x be the number of vanilla, chocolate and coffee cones sold, respectively. Then :

8x+17x+15x=240⇒ 40x=240

X=6.

The number of chocolate sold was 17×6=102.

2:

If the ratio to vanilla to chocolate to coffee cones sold at Frannie’s was 8:17:15 on a particular day, what percent of the cones sold were chocolate?

Answer:

Chocolate cones made up =17/(8+17+15)=17/40=42.5%

Basic arithmetic concepts

A 1:

Which of the following statement is (are) true?

       I.            -2¹⁰>0

    II.            –(-2)¹⁰>0

 III.            2¹⁰-(-2)¹⁰>0

a)    None

b)   | only

c)    ||| only

d)   | and ||| only

e)    All are correct

Answer:

Since 2¹⁰ is positive, -2¹⁰ is negative. (| is false)

Since (-2)¹⁰ is positive, - (-2)¹⁰ is negative. (|| is false)

Since (-2)¹⁰=2¹⁰, 2¹⁰- (-2)¹⁰=0. (||| is false)

Choice a is correct.

A 2:

What is the circumference of a circle whose area is 10 π ?

a)    5π

b)   10π

c)    2π (10)^1/2

d)   π (20)^1/2

Answer:

Since the area of the circle is given by the formula A=πr², then

πr²=10π ⇒ r²=10⇒ r=  (10)^1/2

As  C=2πr, so C=2π (10)^1/2

A 3:

If a=3(x-7) and b=3x-7, what is the value of a-b?

a)    -28

b)   -14

c)    0

d)   3x-14

e)    3x+7

Answer:

a-b=3(x-7)-(3x-7)=3x-21-3x+7=-21+7=-14 (b)

MCQs about percents

1:

What is 45% of 200?

Answer:

X= 45% × 200

X=45/100 × 200 = 90

2:

90 is 45 % of what number?

Answer:

90= 45% × x

Now evaluate the value of x that is 200.

3:

90 is what percent of 200?

Answer:

90=x % of 200

90=x/100 × 200

⇒ x=45.

Important points

A1:                       The product of GCF (greatest common factor) and LCM ( least common factor) of two numbers if equal to the product of two numbers.

LCM×GCF=a×b        (let the numbers are a and b)

A2:                       Every integer greater than 1 that is not a prime can be written as a product of primes. e.g, 12 can be written as 2×2×3

A3:                       Every integer has a finite set of factors (or divisors) and an infinite set of multiples.

A4:                       Some important points about integers:

1.     If two integers are both even or both odd, their sum and difference are even.

2.     If one integer is even and the other is  odd, their sum and difference are odd.

3.     The product of two integers is even unless both of them are odd.

A5:                       Important inequalities for numbers between 0 to 1:

1.     If 0<x<1, and a is positive, then xa<a

2.     If 0<x<a, and m and n are integers with m>n>1, then x^m<xⁿ<x.

3.     If 0<x<1, then squrare root of x>x

4.     If 0<x<1, then 1/x>x. In fact,1/x>1

A6:                       If fraction have the same nominator then , if the denominators are positive, then the fraction with smaller denominator is greater.

A7:                       In solving percent problems, first convert sentences into equation, for this replace

1.     “what” by “x”

2.     “of” by sign of multiply i.e, “×”

3.     “is” by sign of equality i.e, “=”

4.     “Percent” means “per cent” means 1/100   . So replace percent by 1/100   .

A8:                       Use following formula while solving percentage questions:

1.     Original × percentage= apart

This formula  can also be written as:  actual × percent = changed.

A9:                       For any positive number a : a % of 100 is a.

A10:                  For any positive numbers a and b: a% of b = b% of a

A11:                  If a set of objects is divided into two groups in the ratio of a:b, then the first group contains a/(a+b)   of the objects and the second group contains b/(a+b) of the objects.

Basic arithemtic concepts

A 1:

What is the value of ||3|-|-5||   ?

a)    -8

b)   -2

c)    0

d)   2

e)    8

A 2:

How many integers satisfy the inequality |x|< π ?

a)    0

b)   3

c)    4

d)   7

e)    More than 7

A 3:

If the product of 10 numbers is positive, what is the greatest number of them that could be negative ?

a)    0

b)   1

c)    5

d)   9

e)    10

Answer:1

=∣3-5∣=∣-2∣=2     (d)

Answer:2

|x|<π⇒-π<x<π⇒-3.14<x<3.14

There are 7 integers which satisfy this inequality : -3,-2,-1,0,1,2,3.

D choice is correct.

Answer:3

Since the product of 10 numbers is positive, all 10 of the numbers could be negative. So (e) is correct.