Using reasoning to solve problems - HiSet: High School Equivalency Test: Math

Each term after the first is generated from the previous term by incrementing; the increment begins with 4 and increases by 2 with each term generated. Observe:

,

the correct response. This is confirmed by adding 16 to this term:

The second and third terms are generated by adding, then multiplying by, 1, respectively:

This is repeated for each subsequent pair of terms, except that the number is increased by 1 for each pair:

,

the correct response. This can be confirmed by adding 5 to this term:

.

The tank holds 16 gallons total, so when the gas gauge reads half full, there will be

gallons in the tank; this means that the car will have used up 8 gallons. The car gets 20 miles to the gallon, so the car will have traveled

.

The car will travel at a constant speed of 50 miles per hour. We can set and in the rate formula:

.

The car will leave at noon, so it will be between 3 PM and 4 PM when the gas gauge reads half full.

The sequence is formed as follows:

All of the numbers in the sequence can be seen to be those formed only with 0's and 1's. The first two terms are all such one-digit numbers in descending order:

The next two terms are all such two-digit numbers, again in descending order:

The next four terms are all such three-digit numbers, again in descending order:

The next eight terms are all such four-digit numbers, again in descending order:

The missing term is 1101. This is confirmed by the order of other three four-digit numbers given in the sequence.

The sequence is one of sets. Note that in each term, the final element in each subsequent set increases by 1. Also note that each set comprises the set of all factors of its greatest element. Therefore, the missing set is the set that includes 7 and all of its other factors. Since 7 is a prime number, its set of factors is simply , making this the correct response.

This is the sequence of all prime numbers - the natural numbers with exactly two factors, one and itself - in ascending order. After 19, the next two prime numbers are 23 and 27, making this the correct choice.

The tank holds 16 gallons total, so when the gas gauge reads one-fourth full, there will be

gallons in the tank; this means that the car will have used up 12 gallons. The car gets 25 miles to the gallon, so the car will have traveled

.

The car will travel at a constant speed of 65 miles per hour. We can set and in the rate formula:

The car will leave at noon, so the gas gauge will read one fourth full between 4 PM and 5 PM.

One gallon is equivalent to 4 quarts, so 16 gallons can be converted to quarts by multiplying by conversion factor 4:

One quart is equivalent to 2 pints, so pints can be converted to quarts by dividing by conversion factor 2:

Added together:

,

the correct response.

One gallon is equivalent to 4 quarts, so gallons can be converted to quarts by multiplying by conversion factor 4:

One quart is equivalent to 2 pints, so 10 pints can be converted to quarts by dividing by conversion factor 2:

Added together:

,

the correct response.

One meter is equivalent to 100 centimeters, so meters can be converted to centimeters by multiplying by conversion factor 100:

One centimeter is equivalent to 10 millimeters, so 50 millimeters can be converted to centimeters by dividing by conversion factor 10:

Add these quantities:

,

the correct response.

One meter is equivalent to 100 centimeters, so 2 meters can be converted to centimeters by multiplying by conversion factor 100:

One centimeter is equivalent to 10 millimeters, so millimeters can be converted to centimeters by dividing by conversion factor 10:

Add these quantities:

,

the correct answer.

One yard is equivalent to 3 feet, so feet can be converted to yards by dividing by the conversion factor 3:

Also, one yard is equal to 36 inches, so 18 inches can be converted to yards by dividing by the conversion factor 36;

Add these quantities:

Rewriting with the least common denominator, which is 6:

,

the correct choice.

One yard is equivalent to 3 feet, so 2 feet can be converted to yards by dividing by the conversion factor 3:

Also, one yard is equal to 36 inches, so inches can be converted to yards by dividing by the conversion factor 36;

Add these quantities:

Rewriting with the least common denominator, which is 36:

,

the correct choice.

One gram is equivalent to 1,000 milligrams, so milligrams can be converted to grams by dividing by conversion factor 1,000:

One gram is also equivalent to 100 centigrams, so centigrams can be converted to grams by dividing by conversion factor 100:

Add:

Rewrite in terms of lowest common denominator 1,000:

,

the correct response.

The measure of a rectangular plot of land is equal to the product of its dimensions, so the given plot has area

One square mile is equivalent to 640 acres, so multiply this by conversion factor 640:

,

the correct response.

One square mile is equivalent to 640 acres. Therefore, the area of the plot, 72 acres, can be converted to square miles by dividing by conversion factor 640:

If we let and be the north-south and east-west lengths in miles of the rectangular land plot, respectively, then

Substitute for :

Solve for by multiplying both sides by 4:

The east-west length of the plot is mile.

One gallon is equivalent to 4 quarts.

One quart is equivalent to 2 pints.

One pint is equivalent to 2 cups.

Therefore, one gallon is equivalent to cups.

Pouring out one cup leaves 15 cups, the correct response.

528 yards, the sidelength of the square plot, can be converted to square miles as follows:

1,760 yards are equivalent to one mile. Therefore, 528 can be divided by conversion factor 1,760:

The area of the plot is equal to the square of this, or

One square mile is equivalent to 640 acres, so multiply by conversion factor 640:

,

the correct response.

This problem can be solved using a proportion statement.

One meter is equal to 100 centimeters, which in turn is equal to 39.37 inches. Letting be the centimeter equivalent of 30 inches, the proportion statement that relates centimeters and inches can be written

Multiply both sides by 30: