Tom has 7 times as many crayons as Dick and 3 times as many as Harry.
So, the number of crayons Tom has must be a multiple of 21 (i.e., 7 × 3).
Therefore, 140 and 120 are not possible options.
Also, if Tom has 147 crayons, then Dick will have 21 crayons; but, Dick has less than 21 crayons.
So, the maximum number of crayons that Tom can have is 105.

Now, 1001 is divisible by neither 3 nor 5.
So, let's try to divide by 7. We find 1001 / 7 = 143 = 13 × 11.
Since 1001 = 13 × 11 × 7, the smallest prime factor is 7.

The arithmetic mean of n numbers is their sum divided by n.
So, (3 + 5 + 7 + y) / 4 = 100 or 15 + y = 100 × 4 = 400.
Thus, y = 400 − 15 = 385.
Given that 3, 5 and 7 are small numbers and the average 100 is relatively large, it is possible to guess the answer as 385.

Percent of false questions = (3/7) × 100 = 42.86.
So, about 43% of the questions are false.
Alternatively, one could reason out without any calculations as follows.
For every 4 true questions, there are 3 false questions. So, there are fewer false questions than true ones, i.e., less than 50% of the questions are
false. The only option is 43%.

Let b be the side of the square and d the length of its diagonal.
Then, by Pythagoras theorem, d² = b² + b² = 2 b².
Since the diagonal of the larger square is 3 times that of the small square,
the side of the larger square is 3 times the side of the small square.
If b is the side of the small shaded square, then its area = b².
Now, 3b is the side of the larger square and its area = 9b².
Thus, unshaded area = 8b² and
fraction of the area of the larger square unshaded = 8/9.

Percent Commission = (0.40/50) × 100 = 40/50 = 0.8.
Alternatively, 1% of $50 is 50 cents. Since the commission is 40 cents, it is less than 1%. So, the only option is 0.8%.

2% of 7 = (2/100) × 7 = 14/100 = 0.14. So, 2% of 7% = 0.14%.

Now, (2.5/100) z = (75/100) 50. So, z = (75/2.5) 50 = (750/25) 50 = 1500.
Alternatively, 75 is 30 times 2.5. So, z must be 30 times 50.
Of the given options, only 1500 is large enough.
To save time, you can sometimes guess the answer without doing detailed computations.

Let the smallest of the four consecutive odd integers be m. Then,
m + (m + 2) + (m + 4) + (m + 6) = 968 or 4m + 12 = 968.
Thus, m = 956 / 4 = 239.
The four consecutive odd integers are 239, 241, 243 and 245.

Let the car's price in 1998 be x dollars.
Then, a 20% price decrease implies the price in 2000 was (1 − 0.2)x.
So, P=0.8x = 4x/5 or x = 5P/4 = 1.25P.

40% = 40/100 = 4/10 = 2/5, whereas 400/500 = 40/50 = 80/100 = 0.8 = 4/5.

The office requires PE sheets per month.
Therefore, the T sheets of paper will last T/PE months.
If the problem appears confusing, choose an easy set of numbers,
e.g., P = 30; E = 10 and T = 600.
Then, the office requires 300 sheets (i.e., 30 × 10) per month.
Therefore, the 600 sheets of paper will last 2 months (i.e., 600/300).

Adding the equations gives 2x = 15 or x = 7.5. So, x > 5 (II is true).
Substituting x = 7.5 gives z − y = 2.5. So, z > y (I is false).
There are no further conditions on y. So, III is not necessarily true.

Let c be the computer sales in 1999 and p the printer sales in 1999.
Then, the computer sales in 2000 are 1.4c and the printer sales in 2000 are 0.6p.
Ratio of printer sales to computer sales in 2000 = 0.6p/1.4c.
Ratio of printer sales to computer sales in 1999 = p/c.
Dividing the above two ratios, we get 0.6/1.4 = 6/14 = 3/7.

Manufacturing Cost = (d dolls/hour) (c cents/doll)
= cd cents/hour = cd/100 $/hr.
Now, 7 hours and 30 minutes = 7 1/2 hours = 15/2 hours.
Total Manufacturing Cost = (cd/100 $/hr) (15/2 hr) = 15cd/200 $ = 3cd/40 $.

The given inequality can be simplified as follows.
2z² + 5z − 15 − 2z² + 5z − 30 > 0
10z − 45 > 0 or 10z > 45 or z > 4.5

If $45,000 is divided equally among the three, then each will get 5,000. Since the prize money is to be divided unequally, the largest prize will be more than 5,000. The only option is $27,000.

The arithmetic mean of n numbers is their sum divided by n.
So, Sum of five numbers / 5 = −5 or Sum of five numbers = −25.
Now, Sum of three numbers + 50 = −25 or Sum of three numbers = −75.
Thus, Average = Sum of three numbers / 3 = −75/3 = −25.
Here is an alternative strategy:
Given that the average of all the five numbers is negative (−5), their sum is negative. Given that the sum of two of them is 50 (positive), the sum and
average of the other three must be negative. So, the only options are −10 and −25.
Check for −10: Average of five numbers = (50 + 3 (−10))/5 = 4. So, incorrect.
Check for −25: Average of five numbers = (50 + 3 (−25))/5 = −5. So, correct.

Consider one-fourth of the square with half a semicircle as shown in the figure.
Area of one-fourth of the square = 1 × 1 = 1.
Area of half a semicircle = Area of circle / 4 = π(radius)²/4 = π(1)²/4 = π/4.
Area of shaded portion in figure alongside = 1 − π/4.
There are eight such shaded portions in the complete square.
So, total area of shaded region in the complete square = 8 − 2π.

Let Jill be y years old. So, Jack is 7y years old now.
Ten years from now, Jill will be (y + 10) and Jack will be (7y + 10).
So, 7y + 10 = 3 (y + 10) = 3y + 30.
Thus, 4y = 20 or y = 5. Jill is 5 years old and Jack is 35 years old now.