Wiley the official guide for GMAT Episode 1 Part 6 ppt

If 57 percent of the seeds in plot I germinated and 42 percent of the seeds in plot II germinated, what percent of the total number of planted seeds germinated?... Because this was out

end Th e points (0,–1), (1,0), (2,1), and (3,2) are

on segment PQ, and they divide the segment into

three intervals of equal length as shown in the

figure below

Q(3,2) y

1 2

–1 –1

Note that the point (2,1) is twice as far from

P (0,–1) as from Q (3,2) and also that it is 1

A quick look at the answer choices reveals the

expression 2n in answer choice C 2n is a multiple

of 2 and hence must be even

Since only one answer choice can be correct, the

other answer choices need not be checked

However, for completeness:

A n + 1 is odd if n is even and even if n is odd

Th erefore, it is not true that n + 1 must be

even

B n + 2 is even if n is even and odd if n is odd

Th erefore, it is not true that n + 2 must be

even

D 2n + 1 is odd whether n is even or odd

Th erefore, it is not true that 2n + 1 must be

even

E n2 is even if n is even and odd if n is odd

Th erefore, it is not true that n2 must be even

Th e correct answer is C.

41 If 4 is one solution of the equation x2 + 3x + k = 10, where k is a constant, what is the other solution?

(A) –7 (B) –4 (C) –3 (D) 1 (E) 6

If 4 is one solution of the equation, then substitute

4 for x and solve for k.

Using the given pattern, with a = 3, b = 5, c = –2, and d = 4, gives 3 5

2 4

− = (3)(4) – (5)(–2) =

12 + 10 = 22

Th e correct answer is E.

43 The sum 78 + 19 is between

(A) 1

2 and

3 4

Since it is given that x = y, set the expressions for

x and y equal to each other and solve for t

Perform the arithmetic calculations as follows:

Work the problem

47 In a horticultural experiment, 200 seeds were planted

in plot I and 300 were planted in plot II If 57 percent

of the seeds in plot I germinated and 42 percent of the seeds in plot II germinated, what percent of the total number of planted seeds germinated?

Because this was out of 500 seeds planted, the

percent of the total planted that germinated was

Note: Figure not drawn to scale.

48 In the fi gure above, if AB || CE, CE = DE, and y = 45,

Note: Figure not drawn to scale.

Since AB || CE, ∠B and ∠ECD are

corresponding angles and, therefore, have the

same measure Since CE = DE, ΔCED is isosceles

so ∠D and ∠ECD have the same measure Th e

angles of ΔCED have degree measures x, x, and y,

so 2x + y = 180 Since y = 45, 2x + y = 180

2x + 45 = 180 2x = 135

Isolate the variable in the inequalities to

determine the range within which n lies

1 < 5n + 5 < 25

−4 < 5n < 20 subtract 5 from all three values

− 4

5 < n < 4 divide all three values by 5

Th ere are four integers between − 4

integers

Since y is an integer, 23 – 5y is also an integer

Th e task is to fi nd the integer y for which

|23 – 5y| is the least If y ≥ 0, –5y ≤ 0, and

23 – 5y ≤ 23 On the other hand, if y ≤ 0, –5y ≥ 0,

and 23 – 5y ≥ 23 Th erefore, the least possible

value of |23 – 5y| occurs at a nonnegative value of

y From the chart below, it is clear that the least

possible integer value of |23 – 5y| is 2, which

Alternatively, since |23 – 5y| ≥ 0, the minimum

possible real value of |23 – 5y| is 0 The integer

value of y for which |23 – 5y| is least is the integer

closest to the solution of the equation 23 – 5y = 0

Let x represent the people over 21 Th en 3x

represents the number of people 21 or under,

and x + 3x = 4x represents the total population

Th us, the ratio of those 21 or under to the total population is 3

4

34

Th e sum of the measures of angles that form a

straight line equals 180 From this, 2x + 3x = 180

so 5x = 180 and thus x = 36 Th en, because vertical angles are congruent, their measures in degrees are equal Th is can be expressed in the

following equation and solved for y:

2x = y + 30 2(36) = y + 30 substitute 36 for x

72 = y + 30 simplify

42 = y subtract 30 from both sides

Rewrite each radical in the form a b , where a

and b are positive integers and b is as small as

possible, and then add

55 Kelly and Chris packed several boxes with books

If Chris packed 60 percent of the total number of

boxes, what was the ratio of the number of boxes

Kelly packed to the number of boxes Chris packed?

If Chris packed 60 percent of the boxes, then

Kelly packed 100 – 60 = 40 percent of the boxes

Th e ratio of the number of boxes Kelly packed to

the number Chris packed is 40

60

23

?

(A) 1 10

(B) 18

(C) 1 4

(D) 5 4

(E) 25 2

Simplify the expression using approximations

Th e correct answer is B

57 The average (arithmetic mean) of 10, 30, and 50 is

5 more than the average of 20, 40, and

(A) 15 (B) 25 (C) 35 (D) 45 (E) 55

is 30 – 5 = 25 Letting x represent the missing

number, set up the equation for calculating the average for the second set of numbers, and solve

for x

20 + 40 + x

3 = 25

60 + x

3 = 25 simplify

60 + x = 75 multiply both sides by 3

x = 15 subtract 60 from both sides

Th e correct answer is A

y = kx + 3

58 In the equation above, k is a constant If y = 17 when

x = 2, what is the value of y when x = 4 ?

If y = kx + 3 and y = 17 when x = 2, then

59 Each week, Harry is paid x dollars per hour for the fi rst

30 hours and 1.5x dollars for each additional hour

worked that week Each week, James is paid x dollars

per hour for the fi rst 40 hours and 2x dollars for each

additional hour worked that week Last week James

worked a total of 41 hours If Harry and James were

paid the same amount last week, how many hours did

Harry work last week?

Harry’s pay, H, is given by

H = and James’s pay, J, is given by

J =

James worked 41 hours, for which his pay was

40x + 2x(41 – 40) = 42x Harry was paid the same amount as James, so Harry’s pay was also 42x

Th us,

42x = 30x + 1.5x(h – 30) 12x = 1.5x(h – 30)

a 20-day period What percent of the original amount

of water evaporated during this period?

10

× 100 = 0.02 × 100 or 2%

Th e correct answer is D.

61 A glucose solution contains 15 grams of glucose

per 100 cubic centimeters of solution If 45 cubic

centimeters of the solution were poured into an

empty container, how many grams of glucose would

Let x be the number of grams of glucose in the

45 cubic centimeters of solution Th e proportion

comparing the glucose in the 45 cubic centimeters

to the given information about the 15 grams of

glucose in the entire 100 cubic centimeters of

solution can be expressed as x

45

15100

Solving the first equation for y gives y = 70

Substituting this into the second equation gives 2(70) + x = 180

(C) 6 5

(D) 1 3

(E) 3 10

Since 1 km ≈ 0.6 mi = 53mi, divide to fi nd that

What is the value of the account today?

(A) $10,350 (B) $10,395 (C) $10,500 (D) $11,500 (E) $12,705

Arithmetic Percents

Th e first year’s increase of 10 percent can be

expressed as 1.10; the second year’s increase of

5 percent can be expressed as 1.05; and the third

year’s decrease of 10 percent can be expressed as

0.90 Multiply the original value of the account by

each of these yearly changes

10,000(1.10)(1.05)(0.90) = 10,395

Th e correct answer is B

65 A certain fruit stand sold apples for $0.70 each and

bananas for $0.50 each If a customer purchased both

apples and bananas from the stand for a total of

$6.30, what total number of apples and bananas did

the customer purchase?

If each apple sold for $0.70, each banana sold for

$0.50, and the total purchase price was $6.30,

then 0.70x + 0.50y = 6.30, where x and y are

positive integers representing the number of

apples and bananas, respectively, the customer

Since y must be an integer, 9 – x must be divisible

by 5 Furthermore, both x and y must be positive

integers For x = 1, 2, 3, 4, 5, 6, 7, 8, the

corresponding values of 9 – x are 8, 7, 6, 5, 4, 3, 2,

and 1 Only one of these, 5, is divisible by 5

(A) 16 to 15 (B) 9 to 5 (C) 5 to 16 (D) 5 to 4 (E) 4 to 5

If F, S, T, and R represent the number of fi rst,

second, third, and fourth graders, respectively, then the given ratios are: (i) S

R R

67 Two integers will be randomly selected from the sets

above, one integer from set A and one integer from set

B What is the probability that the sum of the two

integers will equal 9 ?

(A) 0.15 (B) 0.20 (C) 0.25 (D) 0.30 (E) 0.33

Arithmetic; Algebra Probability;

Concepts of sets

Th e total number of diff erent pairs of numbers,

one from set A and one from set B is (4)(5) = 20

Of these 20 pairs of numbers, there are 4 possible

pairs that sum to 9: 2 and 7, 3 and 6, 4 and 5, and

5 and 4 Th us, the probability that the sum of the

two integers will be 9 is equal to

Th e correct answer is B

68 At a certain instant in time, the number of cars, N,

traveling on a portion of a certain highway can be

estimated by the formula

N =

where L is the number of lanes in the same direction,

d is the length of the portion of the highway, in feet,

and s is the average speed of the cars, in miles per

hour Based on the formula, what is the estimated

number of cars traveling on a -mile portion of the

highway if the highway has 2 lanes in the same

direction and the average speed of the cars is 40

miles per hour? (5,280 feet = 1 mile)

Substitute L = 2, d = (5,280), and s = 40 into the

given formula and calculate the value for N.

Th e correct answer is D.

400,000 300,000 200,000 100,000 0

1990 1992 1994 1996 1998 2000

year

NUMBER OF SHIPMENTS OF MANUFACTURED HOMES

IN THE UNITED STATES, 1990–2000

69 According to the chart shown, which of the following is closest to the median annual number of shipments of manufactured homes in the United States for the years from 1990 to 2000, inclusive?

(A) 250,000 (B) 280,000 (C) 310,000 (D) 325,000 (E) 340,000

Arithmetic Interpretation of graphs and

Since there are 11 entries in the table and 11 is

an odd number, the median of the numbers of

shipments is the 6th entry when the numbers of

shipments are arranged in order from least to

greatest In order, from least to greatest, the fi rst

6 entries are:

Number of shipments180,000190,000210,000270,000270,000310,000

(E) 4

Since y ≠ 0, it is possible to simplify this equation

and solve for x as follows:

divide both sides by y 3x − 5 = 2 multiply both sides by 2

3x = 7 solve for x

x = 7

3

Th e correct answer is C

71 If x + 5 > 2 and x – 3 < 7, the value of x must be

between which of the following pairs of numbers?

(A) –3 and 10 (B) –3 and 4 (C) 2 and 7 (D) 3 and 4 (E) 3 and 10

Th e lowest value that can be divided evenly by

8 and 12 is their least common multiple (LCM)

Since 8 = 23 and 12 = 22(3), the LCM is

Given that r = 0.345, s = (0.345)2, and t = 0 345 ,

s and t can be expressed in terms of r as r2 and

r

1

2, respectively Because 0 < r < 1, the value of

r x decreases as x increases For example, 1 < 2, but

n, are parked in the lot?

It is given that n is the number of trucks and cars

parked in the lot and the number of cars is 1

4 the

number of trucks, so if t represents the number

of trucks and c represents the number of cars,

greatest number of members who could vote against

the resolution and still have it pass?

(A) 19 (B) 17 (C) 16 (D) 14 (E) 13

If at least 2

3 of the members must vote in favor of

a resolution, then no more than 1

3 of the members can be voting against it On this 40-member committee, 1

3 (40) = 13

1

3 , which means that

no more than 13 members can vote against the resolution and still have it pass

Th e correct answer is E

76 In the Johnsons’ monthly budget, the dollar amounts allocated to household expenses, food, and

miscellaneous items are in the ratio 5:2:1, respectively

If the total amount allocated to these three categories

is $1,800, what is the amount allocated to food?

(A) $900 (B) $720 (C) $675 (D) $450 (E) $225

Since the ratio is 5:2:1, let 5x be the money allocated to household expenses, 2x be the money allocated to food, and 1x be the money allocated

to miscellaneous items Th e given information can then be expressed in the following equation and

solved for x

5x + 2x + 1x = $1,800

8x = $1,800 combine like terms

x = $225 divide both sides by 8

Th e money allocated to food is

2x = 2($225) = $450

Th e correct answer is D

77 There are 4 more women than men on Centerville’s

board of education If there are 10 members on the

board, how many are women?

Let m be the number of men on the board and w

be the number of women on the board According

to the problem,

w = m + 4 because there are 4 more women

than men and

w + m = 10 because the board has a total of

10 members

Substituting m + 4 for w in the second equation

gives:

m + m + 4 = 10

2m + 4 = 10 combine like terms

2m = 6 subtract 4 from both sides

m = 3 divide both sides by 2

Using the fi rst equation, w = m + 4 = 3 + 4 = 7

women on the board

Th is problem can also be solved without algebra

by listing the (m,w) possibilities for w = m + 4

Th ese possibilities are (0,4), (1,5), (2,6), (3,7), etc.,

and hence the pair in which m + w = 10 is (3,7).

Th e correct answer is D.

78 Leona bought a 1-year, $10,000 certificate of deposit

that paid interest at an annual rate of 8 percent

compounded semiannually What was the total amount

of interest paid on this certificate at maturity?

Using the formula A P r

n

nt

= (1+ ) , where A is the amount of money after t (1 year), P is the principal amount invested ($10,000), r is the annual interest rate (0.08), and n is the number

of times compounding occurs annually (2), the given information can be expressed as follows

and solved for A :

Th us, since A is the final value of the certificate,

the amount of interest paid at maturity is

To make the calculations less tedious, convert the decimals to whole numbers times powers of 10 as follows:

80 Machine A produces bolts at a uniform rate of 120

every 40 seconds, and Machine B produces bolts at

a uniform rate of 100 every 20 seconds If the two

machines run simultaneously, how many seconds will

it take for them to produce a total of 200 bolts?

Determine the production rates for each machine

separately, and then calculate their production

Combined rate = 3 + 5 = 8 bolts per second

Build an equation with s = the number of seconds

it takes to produce 200 bolts

8s = 200 (rate)(time) = amount produced

(A) 12.0 (B) 12.1 (C) 12.2 (D) 12.3 (E) 12.4

Second-degree equations

Let f be the factor by which the amount of

bacteria present increased every 3 hours

Th en, from the table, 10.0f = x and xf = 14.4

Substituting 10.0f for x in the second equation

gives

(10.0f )f = 14.4 10.0f 2 = 14.4

f 2 = 1.44

f = 1.2 and then x = 10.0(1.2) = 12.0.

Th e easiest and quickest way to do this problem is

to choose an integer greater than 6, such as 7, and

eliminate answer choices in which the value of the

expression is not divisible by 3:

Choose another integer greater than 6, such as 8,

and test the remaining answer choices:

divisible by 3, so E can be eliminated

Th us, A is the only answer choice that has not

been eliminated

For the more mathematically inclined, if n is

divisible by 3, then the expression in each answer

choice is divisible by 3 Assume, then, that n is

not divisible by 3 If the remainder when n is

divided by 3 is 1, then n = 3q + 1 for some integer

q All of the expressions n – 4, n – 1, n + 2, and

n + 5 are divisible by 3 [i.e., n – 4 = 3q – 3 =

3(q – 1), n – 1 = 3q, n + 2 = 3q + 3 = 3(q + 1),

n + 5 = 3q + 6 = 3(q + 2)], and none of the

expressions n – 6, n – 5, n – 2, n + 1, n + 3, and

n + 4 is divisible by 3 Th erefore, if the remainder

when n is divided by 3 is 1, only the expressions

in answer choices A, B, and E are divisible by 3

On the other hand, if the remainder when n is

divided by 3 is 2, then n = 3q + 2 for some integer

q All of the expressions n – 5, n – 2, n + 1, and

n + 4 are d ivisible by 3 [i.e., n – 5 = 3q – 3 =

3(q – 1), n – 2 = 3q, n + 1 = 3q + 3 = 3(q + 1),

n + 4 = 3q + 6 = 3(q + 2)], and none of the

expressions n – 6, n – 4, n – 1, n + 2, n + 3, and

n + 5 is divisible by 3 Th erefore, if the remainder

when n is divided by 3 is 2, only the expressions

in answer choices A, C, and D are divisible by 3

Only the expression in answer choice A is

divisible by 3 regardless of whether n is divisible

by 3, has a remainder of 1 when divided by 3, or has a remainder of 2 when divided by 3

(A) $3.00 (B) $3.75 (C) $4.50 (D) $5.00 (E) $5.50

Th e total cost to produce 20,000 tools is

$10,000 + $3(20,000) = $70,000 Th e revenue resulting from the sale of 20,000 tools is

Th e correct answer is C.

84 A dealer originally bought 100 identical batteries at

a total cost of q dollars If each battery was sold at

50 percent above the original cost per battery, then,

in terms of q, for how many dollars was each battery

Algebra Factoring and Simplifying algebraic

expressions

Since 100 batteries cost q dollars, division by 100

shows that 1 battery costs 100 dollars Thq en, since

the selling price is 50 percent above the original

cost per battery, the selling price of each battery

can be expressed as

Th e correct answer is A.

85 In an increasing sequence of 10 consecutive integers,

the sum of the fi rst 5 integers is 560 What is the sum

of the last 5 integers in the sequence?

Let the fi rst 5 consecutive integers be represented

Th e fi rst integer in the sequence is 110, so the next

integers are 111, 112, 113, and 114 From this, the

last 5 integers in the sequence, and thus their sum,

can be determined Th e sum of the 6th, 7th, 8th,

9th, and 10th integers is

115 + 116 + 117 + 118 + 119 = 585

Th is problem can also be solved without algebra:

Th e sum of the last 5 integers exceeds the sum

of the fi rst 5 integers by 1 + 3 + 5 + 7 + 9 = 25

because the 6th integer exceeds the 5th integer by

1, the 7th integer exceeds the 4th integer by 3, etc

Th e correct answer is A.

86 Machine A produces 100 parts twice as fast as

Machine B does Machine B produces 100 parts in

40 minutes If each machine produces parts at a

constant rate, how many parts does Machine A produce in 6 minutes?

(A) 30 (B) 25 (C) 20 (D) 15 (E) 7.5

If Machine A produces the parts twice as fast as Machine B does, then Machine A requires half

as much time as Machine B does to produce

100 parts So, if Machine B takes 40 minutes for the job, Machine A takes 20 minutes for the job

Th is is a rate of 100 parts

20 minutes = 5 parts per minute

At this rate, in 6 minutes Machine A will produce 5(6) = 30 parts

Th e correct answer is A

87 A necklace is made by stringing N individual beads

together in the repeating pattern red bead, green bead, white bead, blue bead, and yellow bead If the necklace design begins with a red bead and ends with

a white bead, then N could equal

(A) 16 (B) 32 (C) 4 1 (D) 54 (E) 68

Th e bead pattern repeats after every fifth bead

Since the first bead in this design (or the first in the pattern) is red and the last bead in this design (or third in the pattern) is white, the number of beads in this design is 3 more than some multiple

of 5 Th is can be expressed as 5n + 3, where n is

an integer Test each of the answer choices to determine which is a multiple of 5 plus a value

of 3 Of the options, only 68 = 5(13) + 3 can be

written in the form 5n + 3

Th e correct answer is E

88 In the xy-coordinate system, if (a,b) and (a + 3,b + k)

are two points on the line defined by the equation

Substituting the given coordinates for x and y in

the equation x = 3y – 7 yields

a = 3b − 7

a + 3 = 3(b + k) − 7

Th en substitute 3b – 7 for a in second equation,

and solve for k

3b − 7 + 3 = 3b + 3k − 7

3b – 4 = 3b + 3k – 7 combine like terms

3 = 3k subtract 3b from and

add 7 to both sides

1 = k divide both sides by 3

Th e correct answer is D

89 If s is the product of the integers from 100 to 200,

inclusive, and t is the product of the integers from 100

to 201, inclusive, what is in terms of t ?

= =

Let J represent Jake’s weight and S represent his

sister’s weight Th en J – 8 = 2S and J + S = 278

Solve the second equation for S and get

S = 278 – J Substituting the expression for

S into the fi rst equation gives

J – 8 = 2(278 – J)

J – 8 = 556 – 2J

J + 2J = 556 + 8 3J = 564

J = 188

Th e correct answer is E.

91 A certain store sells all maps at one price and all

books at another price On Monday the store sold

12 maps and 10 books for a total of $38.00, and on

Tuesday the store sold 20 maps and 15 books for a

total of $60.00 At this store, how much less does a

map sell for than a book?

Let m represent the price of each map and b

represent the price of each book Th en the given

information can be represented by the system

Multiplying the top equation

by –3

the two equations gives 2m = 3 or m = 1.5

Th us, each map sells for $1.50 Th en,

12(1.50) + 10b = 38

18 + 10b= 38

10b = 20

b = 2

So, each book sells for $2.00 and each map sells

for $1.50, which is $2.00 – $1.50 = $0.50 less than

each book

Th e correct answer is B.

92 A store reported total sales of $385 million for

February of this year If the total sales for the same

month last year was $320 million, approximately what

was the percent increase in sales?

93 If the median of the numbers in list I above is equal to the median of the numbers in list II above, what is the

value of x ?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

Arithmetic Statistics

Since list I has an even number of numbers, the median of list I is the average of the middle two numbers, so 6 + 8

2 = 7 is the median of list I

Since list II has an odd number of numbers, the median of list II will be the middle number when the five numbers are put in ascending order

Since the median of list II must be 7 (the median

of list I) and since 7 is not in list II, then x = 7

Th e correct answer is B

94 In a certain city, 60 percent of the registered voters

are Democrats and the rest are Republicans In a

mayoral race, if 75 percent of the registered voters

who are Democrats and 20 percent of the registered

voters who are Republicans are expected to vote for

Candidate A, what percent of the registered voters are

expected to vote for Candidate A ?

Letting v be the number of registered voters in

the city, then the information that 60% of the

registered voters are Democrats can be expressed

as 0.60v From this, it can be stated that

1.00v – 0.60v = 0.40v are Republicans Th e

percentage of Democrats and the percentage

of Republicans who are expected to vote

for Candidate A can then be expressed as

(0.75)(0.60v) + (0.20)(0.40v) Simplify the

expression to determine the total percentage

of voters expected to vote for Candidate A

Perform the operations in the correct order, using least common denominators when adding or subtracting fractions:

(A) 16 (B) 72 (C) 112 (D) 128 (E) 142

Th e mass ratio of oxygen to water is

= 16

16+2 =

8

9 Th erefore, if

x is the number of grams of oxygen in 144 grams

of water, it follows that x

Setting each factor equal to 0, it can be seen that

the solution set to the fi rst equation is

and the solution set to the second equation is

Th erefore, –1

2 is the solution to both equations

Th e correct answer is B.

98 On a scale that measures the intensity of a certain

phenomenon, a reading of n + 1 corresponds to an

intensity that is 10 times the intensity corresponding

to a reading of n On that scale, the intensity

corresponding to a reading of 8 is how many times

as great as the intensity corresponding to a reading

Since 8 can be obtained from 3 by “adding 1” fi ve

times, the intensity reading is greater by a factor

of (10)(10)(10)(10)(10) = 105

Th e correct answer is C.

99 For the positive numbers, n, n + 1, n + 2, n + 4, and

n + 8, the mean is how much greater than the median?

(A) 0 (B) 1 (C) n + 1

(D) n + 2

(E) n + 3

Algebra Statistics

Since the fi ve positive numbers n, n + 1, n + 2,

n + 4, and n + 8 are in ascending order, the median is the third number, which is n + 2 Th e mean of the fi ve numbers is

than the median

Th e correct answer is B.

100 If T = 5

9 (K – 32), and if T = 290, then K =

(A) 1,7389 (B) 322

(C) 490

(D) 554

(E) 2,8985

Algebra First-degree equations

Substitute 290 for T in the equation, and solve

101 The water from one outlet, flowing at a constant rate,

can fill a swimming pool in 9 hours The water from a

second outlet, flowing at a constant rate, can fill the

same pool in 5 hours If both outlets are used at the

same time, approximately what is the number of hours

required to fill the pool?

Th e first outlet can fill the pool at a rate of 1

9

of the pool per hour, and the second can fill

the pool at a rate of 1

5 of the pool per hour

Together, they can fill the pool at a rate of

of the pool per hour

Th us, when both outlets are used at the same

time, they fill the pool in hours

Th e correct answer is D

102 If a square mirror has a 20-inch diagonal, what is the approximate perimeter of the mirror, in inches?

(A) 40 (B) 60 (C) 80 (D) 100 (E) 120

Let x be the length of one of the sides of the

it is to 1,600 or 6,400

Th e correct answer is B.

103 The present ratio of students to teachers at a certain school is 30 to 1 If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to