Hard Exponents Question

Decimal Computation with [ Number Properties ] GMAT, GRE

$\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}=$




(A) 10^(-8)
(B) 3*10^(-8)
(C) 3*10^(-4)
(D) 2*10^(-4)
(E) 10^(-4)

Integer Digits [ Combinations ] GMAT, GRE

How many positive integers less than or equal to 2017 contain the digit 0?

(A) 469
(B) 471
(C) 475
(D) 478
(E) 481

3/8 of all students at a school are in all three of the following clubs: Albanian, Bardic, and Chess. 1/2 of all students are in Albanian, 5/8 are in Bardic, and 3/4 are in Chess  If every student is in at least one club, what fraction of the student body is in exactly 2 clubs?

(A) 1/8
(B) 1/4
(C) 3/8
(D) 1/2
(E) 5/8

Simple Interest GMAT, GRE Problem Solving Guide

All interest referred to below is simple interest. The annual amount of interest paid on an investment is found by multiplying the amount invested (the principal) by the percent of interest (the rate).

Example:

Sam invested some of his money in a bank paying 4% interest annually and a second amount, $500 less than the first, in a bank paying 6% interest. If her annual income from both investments was $50, how much money did Sam invest at 6%?

Solution:

As this is a non-ratio algebraic problem, we can handle this with a Distance = Rate x Time chart. Only we'll need an analogy for our names. Let's use: 

PRINCIPAL · RATE = INTEREST INCOME

Let's put it together algebraically.

x = amount invested at 4%x – 500 = amount invested at 6%
.04x = annual interest from 4% investment
.06(x – 500) = annual interest from 6% investment .04x + .06(x – 500) = 50

Multiply by 100 to remove decimals.

4x + 6(x − 500)= 5000 4x + 6x − 3000 = 5000 10x = 8000
x = 800x − 500 = 300

Sam invested $300 at 6%.

Let's try another using our matrix method

Jill invested $7200, part at 4% and the rest at 5%. If the annual income from both investments was the same, find her total annual income from these investments.

1. (A)  $160
2. (B)  $320
3. (C)  $4000
4. (D)  $3200
5. (E)  $1200
   
   

Let x = amount invested at 4%

7200 − x = amount invested at 5% .04x =.05(7200-x)

Multiply by 100 to eliminate decimals.

4x = 5(7200 − x)4x = 36,000 − 5x9x = 36,000
x = 4000
Her income is .04(4000) + .05(3200). 

$320.

GMAT, GRE Factors, Primes, Functions and [ Number Properties ] Problem

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) +1, then p is?

A. Between 2 and 20
B. Between 10 and 20
C. Between 20 and 30
D. Between 30 and 40
E. Greater than 40 

Medium GMAT, GRE [ Rate Time Distance ] Problem

On a path, a hiker walking at a constant rate of 4 miles per hour is passed by a cyclist traveling in the same direction along the same path at a constant rate of 20 miles per hour. the cyclist stops & waits for the hiker 5 min after passing him while the hiker continues to walk at his constant rate. how many minutes must the cyclist wait until the hiker catches up

A. 6 2/3
B. 15
C. 20
D. 25
E. 26 2/3

How to handle Mixture word problems on the GMAT, GRE

Solving mixture problems follows the same method we like to do for all rate x time = Distance questions. We make a matrix for r x t = D on the top but to fill it in we'll make an analogy. What are our terms instead of r and t? Look by the numbers in the question. In the question below we have 36 and next to it is an amount of money, so one of the terms will be price. Another value is 300 and next to it is a weight, so the other term is number of pounds. Does it matter which comes first? Well, let me ask you: what's 3 x 4? And what's 4 x 3? When we multiply across our matrix to get the "D" or Total value, the order in which we do it doesn't matter. Let's look at our example.

Example:

Mr. Thomson wants to mix candy worth 36 cents a pound with candy worth 52 cents a pound to make 300 pounds of a mixture worth 40 cents a pound. How many pounds of the more expensive candy should he use?

Solution:

Let's set up our r x t = D matrix using our analogie values of pounds, price and total.



The value of the more expensive candy plus the value of the less expensive candy must be equal to the value of the mixture. Almost all mixture problems derive their equation from adding the final column (The "D" or "Distance" value) in the chart. In this case, we changed our Distance to read Total Value.

52x + 36(300 – x) = 12000

Notice that all values were computed in cents to avoid decimals. As long as you multiply all the values of an equation you can manipulate them however you want. 

52x+10,800−36x =12,000 16x = 1200
x = 75

He should use 75 pounds of the more expensive candy.

700 Level GMAT [ Work Rate Ratio ] Problem

Twenty-four men can complete a work in sixteen days.Thirty-two women can complete the same work in twenty-four days.Sixteen men and sixteen women started working for twelve days.How many more men are to be added to complete the work remaining work in 2 days?

A .16
B. 24
C. 36
D. 48
E. 54

Distance = rate x time

For our purposes: Work = Rate (of work for each man person) * (total time)

1 Man 1 Dday work=1/(16*24)
1 Woman 1 D work=1/(24*32)

16 M 1 D work=1/24=2/48
16 W 1 D work=1/48

16 M&W 1 D work=3/48=1/16
16 M&W 12 D work=12/16

The remaining work is 4/16 or 1/4 
this work is to be completed by the additional men

1 M 1 D work = 1/(16*24)
1 M 2 D work = 1/(16*12)
x M 2 D work = x/(16*12) 
this is set to be equal to 1/4

x/(16*12) =1 /4
x = 48

D 

GMAT, GRE [ Remainders ]

${}^{}$What is the remainder when 11^12 is divided by 12 ?





Let's look at the cycle of remainders: 

What is the remainder of 7 divided by 4? Ans = 3
What is the remainder of 6 divided by 4? Ans = 2
What is the remainder of 5 divided by 4? Ans = 1
What is the remainder of 4 divided by 4? Ans = 0
What is the remainder of 3 divided by 4? Ans = -1

(11^1)/12 gives a remainder of 11 or -1

(Remainder of 11/12)^12gives us (-1)^12
which is 1

A straight picket fence is composed of x pickets each of which is 1/2 inch wide. If there are 6 inches of space between each pair of pickets, which of the following represents the length of fence in feet?

A. 13x/2
B. 13x/2 - 6
C. 13x/24
D. (13x+1)/24
E. (13x-12)/24

GMAT, GRE [Inequalities] With Squares

Which of the following describes all values of x for which 1–x^2 ≥ 0?

(A) x ≥ 1
(B) x ≤ –1
(C) 0 ≤ x ≤ 1
(D) x ≤ –1 or x ≥ 1
(E) –1 ≤ x ≤ 1

Range With [Rate] Gmat, Gre Quantitative Problem

On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallon. The actual number of miles per gallon that Cindy's car got on this trip must have been between 

A. 290/12.5 and 290/11.5 
B. 295/12 and 285/11.5 
C. 285/12 and 295/12 
D. 285/12.5 and 295/11.5 
E. 295/12.5 and 285/11.5 

Cindy drove her car 290 miles, rounded to the nearest 10 miles --> 285≤m<295$285\le m<295$;
Used 12 gallons of gasoline, rounded to the nearest gallon --> 11.5≤g<12.5$11.5\le g<12.5$;

Minimum Miles per gallon, m/g --> 28512.5<mg<29511.5$\frac{285}{12.5}<\frac{m}{g}<\frac{295}{11.5}$ 

D

Powers and Remainders [Number Properties] GMAT, GRE Quantitative Problem

z is a positive integer and multiple of 2; p = 4z, what is the remainder when p is divided by 10?

1. A)  10
2. B)  6
3. C)  4
4. D)  0
5. E)  It Cannot Be Determined
   
   
   
   
1. Every time, all the time, the remainder when an integer is divided by 10 is simply the units digit of that integer. To help see this, consider the following examples:
   14/10 is 1 with a remainder of 4
   5/10 is 0 with a remainder of 5
   105/10 is 10 with a remainder of 5
2. Also, z is a positive integer and is a multiple of 2. So, z must be a positive even integer.
3. We can rephrase the question to: "what is the units digit of 4 when raised to an even positive integer power?"
4. All integers raised to consecutive integer powers follow a repeating pattern. The units digit of 4 raised to an integer follows a specific repeating pattern:
   41 = 4
   42 = 16
   43 = 64
   44 = 256 
5. So we can say that4(odd number) --> units digit of 4 and 4(even number) --> units digit of 6
6. Since z must be an even integer, the units digit of p=4z will always be 6. Consequently, the remainder when p=4z is divided by 10 will always be 6.
   here are some examples:
   z=2 --> p=4z=16 --> p/10 = 1 with a remainder of 6
   z=4 --> p=4z=256 --> p/10 = 25 with a remainder of 6
   z=6 --> p=4z=4096 --> p/10 = 409 with a remainder of 6
   z=8 --> p=4z=65536 --> p/10 = 6553 with a remainder of 6

B

GMAT, GRE Units Digit With Exponents

What is the units digit of 615 - 74 - 93?
A) 8 
B) 7 
C) 6 
D) 5 
E) 4

Answer

1. The units digit of any integer raised to a power will follow a pattern. Let's see how powers of 6 play out.
   61 = 6 --> units digit of 6
   62 = 36 --> units digit of 6
   63 = 216 --> units digit of 6
   64 = 1,296 --> units digit of 6
   
2. Let's apply the same process to powers of 7.
   71 = 7 --> units digit of 7
   72 = 49 --> units digit of 9
   73 = 343 --> units digit of 3
   74 --> units digit of 1
   75 --> units digit of 7
   76 --> units digit of 9
   77 --> units digit of 3
   78 --> units digit of 1
   
3. The units digit of 9 raised to an integer exponent follows a definitive pattern as well.
   91 = 9 --> units digit of 9
   92 = 81 --> units digit of 1
   93 --> units digit of 9
   94 --> units digit of 1
   95 --> units digit of 9
   
4. Thus far you know that the units digit of 615 – 74 – 93 = units digit of 6 – units digit of 1 – units digit of 9.
5. Simplify the first two terms of this expression: units digit of 6 – units digit of 1 = units digit of 5
6. So here's where we're at now: units digit of 5 - units digit of 9.
7. Hard Part: Some will think that since 5-9=-4, so units digit of the entire expression will be 4. But this fails to consider that the left term could be larger than the right, resulting in a units digit of 6. For example:
   15-9=6 {left term is larger}
   155-99=56 {left term is larger}
   155-999=-844 {right term is larger}
   15-99=-84 {right term is larger}
8. The crucial question in determining whether the units digit of the final expression is a 6 or a 4 is whether the left expression is larger than the right. In other words, "is (615 - 74) greater than 93?". I like to think of a negative in a units digit equation as its distance from 10. In this case 10-4 = 6.
9. To be sure we're right, take an approximate guess at the value of each term. You know that 93 will be less than 1000, which is 103, so the question of whether the units digit is 6 or 4 really rests on whether 615 - 74 is greater than 1000 (in which case the left term will be larger than the right term and the units digit will be 6) or whether 615 - 74 is less than a thousand (in which case the right term will be larger than the left term and the units digit will be 4).
10. The test does not require long tedious calculations and these are not necessary here. It should be rather clear that 615 - 74 is greater than 1000, in which case the units digit will be 6, not 4.
11. Units digit of 5 - units digit of 9 = units digit of 6 since the left term (i.e., the one with a 5) is larger than the right term. The final answer is a units digit of 6.
12. Answer choice C .

Groups of Groups: GMAT, GRE, SAT problem solving

At Tufton University, 40% of all students are members of both a chess club and a swim team. If 20% of members of the swim team are not members of the chess club, what percentage of all Daifu students are members of the swim team?

A. 20%
B. 30%
C. 40%
D. 50%
E. 60%

Answer:

____________________________________________

Total Students = T
Swim Club Only = S
Chess Club Only = C
Both Swim and Chess = B which is 40% of T = 0.4 T
And C= 0.8 S

____________________________________________

Equations :
S + C+ B = T
S + 0.8S + 0.4T = T
1.8S = 0.6T
So 1.8/0.6 = 30% i.e. 30 % of total are on the swim team

____________________________________________