Quantitative reasoning - HiSET

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Question

Give the missing number in the sequence:

$12, 18, 24, 30, 36, 42, $\underline{; ; ; ; ; ; }$, 54...$

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Answer

Each term in the sequence is generated from the previous term by adding 6,as follows:

$\textbf{12}+ 6 = \textbf{18 }$

$\textbf{18}+ 6 = \textbf{24 }$

$\textbf{24}+ 6 = \textbf{30 }$

$\textbf{30}+ 6 = \textbf{36 }$

$\textbf{36}+ 6 = \textbf{42 }$

$\textbf{42}+ 6 = \textbf{{\color{Red} 48}}$

The correct response is 48. This is confirmed by adding 6 again:

$\textbf{48}+ 6 = \textbf{54 }$

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Question

Give the missing number in the sequence:

$15, 19, 25, 33, 43, 55, $\underline{; ; ; ; ; ; }$, 85,...$

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Answer

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:

$\textbf{15} + 4 = \textbf{19 }$

$\textbf{19} + 6 = \textbf{25 }$

$\textbf{25} + 8= \textbf{33 }$

$\textbf{33} + 10 = \textbf{43 }$

$\textbf{43} + 12 = \textbf{55 }$

$\textbf{55} + 14 = \textbf{{\color{Red} 69} }$ ,

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

$\textbf{69} + 16 = \textbf{85 }$

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Question

Give the missing number in the sequence:

$5, 6, 6, 8, 16, 19, 57, 61 , $\underline{; ; ; ; ; ; }$, 249, ...$

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Answer

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

$\textbf{5}+ 1= \textbf{6}$

$\textbf{6} \times 1= \textbf{6}$

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

$\textbf{6}+ 2= \textbf{8}$

$\textbf{8} \times 2= \textbf{16}$

$\textbf{16}+ 3= \textbf{19}$

$\textbf{19} \times 3= \textbf{57 }$

$\textbf{57}+ 4= \textbf{61}$

$\textbf{61} \times 4= \textbf{{\color{Red} 244}}$ ,

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

$\textbf{244}+ 5 = \textbf{249}$ .

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Question

A car which gets 20 miles per gallon travels at a constant speed of 50 miles per hour. If the car's tank, which holds 16 gallons of gasoline, was filled at noon, when should the gas gauge read half full? (Assume no stops)

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Answer

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

$$\frac{1}{2}$ \times 16 = 8$

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

$20 \textup{ mi/gal } \times 8 \textup{ gal} = 160 \textup{ mi}$ .

The car will travel at a constant speed of 50 miles per hour. We can set $r = 50 \textup{ mi /hr}$ and $d = 160 \textup{ mi }$ in the rate formula:

$t = $\frac{d}{r}$ = $\frac{160 \textup{ mi }$}{50 \textup{ mi /hr}} = 3 $\frac{1}{5}$\textup{ hr}$ .

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

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Question

Give the next number in the sequence:

$1, 0, 11,10, 111, 110, 101, 100, 1111, 1110, $\underline{ ; ; ; ; ; ; ; ; }$, 1100,...$

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Answer

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:

$0, 1 \rightarrow \textbf{1, 0}$

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

$10, 11 \rightarrow \textbf{11, 10 }$

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

$100, 101, 110, 111 \rightarrow \textbf{ 111, 110, 101, 100}$

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

$1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 \rightarrow \textbf{1111, 1110}, \textbf{{\color{Red} 1101}}, \textbf{1100}, 1011,1010, 1001, 1000$

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

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Question

Give the next two terms in the sequence:

$2, 3, 5, 7, 11, 13, 17, 19, $\underline{; ; ; ; ; ; }$, $\underline{; ; ; ; ; ; }$, ...$

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Answer

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.

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Question

A car which gets 25 miles per gallon travels at a constant speed of 65 miles per hour. If the car's tank, which holds 16 gallons of gasoline, was filled at noon, when should the gas gauge read one-fourth full? (Assume no stops)

Tap to reveal answer

Answer

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

$$\frac{1}{4}$ \times 16 = 4$

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

$25 \textup{ mi/gal } \times 12 \textup{ gal} = 300 \textup{ mi}$ .

The car will travel at a constant speed of 65 miles per hour. We can set $r = 65 \textup{ mi /hr}$ and $d = 30 0 \textup{ mi }$ in the rate formula:

$t = $\frac{d}{r}$ = $\frac{300 \textup{ mi }$}{65 \textup{ mi /hr}}\approx 4.6 \textup{ hr}$

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

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Question

Give the missing term in the sequence:

$\left \{ 1 \right \}$

$\left \{ 1,2 \right \}$

$\left \{ 1,3 \right \}$

$\left \{ 1,2,4 \right \}$

$\left \{ 1,5\right \}$

$\left \{ 1,2, 3 , 6 \right \}$

$\left \{ 1,2,4, 8 \right \}$

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Answer

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 $\left \{ 1,7 \right \}$ , making this the correct response.

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Question

Give the missing number in the sequence:

$12, 18, 24, 30, 36, 42, $\underline{; ; ; ; ; ; }$, 54...$

Tap to reveal answer

Answer

Each term in the sequence is generated from the previous term by adding 6,as follows:

$\textbf{12}+ 6 = \textbf{18 }$

$\textbf{18}+ 6 = \textbf{24 }$

$\textbf{24}+ 6 = \textbf{30 }$

$\textbf{30}+ 6 = \textbf{36 }$

$\textbf{36}+ 6 = \textbf{42 }$

$\textbf{42}+ 6 = \textbf{{\color{Red} 48}}$

The correct response is 48. This is confirmed by adding 6 again:

$\textbf{48}+ 6 = \textbf{54 }$

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Question

Give the missing number in the sequence:

$15, 19, 25, 33, 43, 55, $\underline{; ; ; ; ; ; }$, 85,...$

Tap to reveal answer

Answer

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:

$\textbf{15} + 4 = \textbf{19 }$

$\textbf{19} + 6 = \textbf{25 }$

$\textbf{25} + 8= \textbf{33 }$

$\textbf{33} + 10 = \textbf{43 }$

$\textbf{43} + 12 = \textbf{55 }$

$\textbf{55} + 14 = \textbf{{\color{Red} 69} }$ ,

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

$\textbf{69} + 16 = \textbf{85 }$

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Question

Give the missing number in the sequence:

$5, 6, 6, 8, 16, 19, 57, 61 , $\underline{; ; ; ; ; ; }$, 249, ...$

Tap to reveal answer

Answer

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

$\textbf{5}+ 1= \textbf{6}$

$\textbf{6} \times 1= \textbf{6}$

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

$\textbf{6}+ 2= \textbf{8}$

$\textbf{8} \times 2= \textbf{16}$

$\textbf{16}+ 3= \textbf{19}$

$\textbf{19} \times 3= \textbf{57 }$

$\textbf{57}+ 4= \textbf{61}$

$\textbf{61} \times 4= \textbf{{\color{Red} 244}}$ ,

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

$\textbf{244}+ 5 = \textbf{249}$ .

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Question

A car which gets 20 miles per gallon travels at a constant speed of 50 miles per hour. If the car's tank, which holds 16 gallons of gasoline, was filled at noon, when should the gas gauge read half full? (Assume no stops)

Tap to reveal answer

Answer

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

$$\frac{1}{2}$ \times 16 = 8$

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

$20 \textup{ mi/gal } \times 8 \textup{ gal} = 160 \textup{ mi}$ .

The car will travel at a constant speed of 50 miles per hour. We can set $r = 50 \textup{ mi /hr}$ and $d = 160 \textup{ mi }$ in the rate formula:

$t = $\frac{d}{r}$ = $\frac{160 \textup{ mi }$}{50 \textup{ mi /hr}} = 3 $\frac{1}{5}$\textup{ hr}$ .

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

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Question

Give the next number in the sequence:

$1, 0, 11,10, 111, 110, 101, 100, 1111, 1110, $\underline{ ; ; ; ; ; ; ; ; }$, 1100,...$

Tap to reveal answer

Answer

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:

$0, 1 \rightarrow \textbf{1, 0}$

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

$10, 11 \rightarrow \textbf{11, 10 }$

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

$100, 101, 110, 111 \rightarrow \textbf{ 111, 110, 101, 100}$

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

$1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 \rightarrow \textbf{1111, 1110}, \textbf{{\color{Red} 1101}}, \textbf{1100}, 1011,1010, 1001, 1000$

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

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Question

Give the next two terms in the sequence:

$2, 3, 5, 7, 11, 13, 17, 19, $\underline{; ; ; ; ; ; }$, $\underline{; ; ; ; ; ; }$, ...$

Tap to reveal answer

Answer

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.

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Question

A car which gets 25 miles per gallon travels at a constant speed of 65 miles per hour. If the car's tank, which holds 16 gallons of gasoline, was filled at noon, when should the gas gauge read one-fourth full? (Assume no stops)

Tap to reveal answer

Answer

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

$$\frac{1}{4}$ \times 16 = 4$

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

$25 \textup{ mi/gal } \times 12 \textup{ gal} = 300 \textup{ mi}$ .

The car will travel at a constant speed of 65 miles per hour. We can set $r = 65 \textup{ mi /hr}$ and $d = 30 0 \textup{ mi }$ in the rate formula:

$t = $\frac{d}{r}$ = $\frac{300 \textup{ mi }$}{65 \textup{ mi /hr}}\approx 4.6 \textup{ hr}$

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

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Question

Give the missing term in the sequence:

$\left \{ 1 \right \}$

$\left \{ 1,2 \right \}$

$\left \{ 1,3 \right \}$

$\left \{ 1,2,4 \right \}$

$\left \{ 1,5\right \}$

$\left \{ 1,2, 3 , 6 \right \}$

$\left \{ 1,2,4, 8 \right \}$

Tap to reveal answer

Answer

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 $\left \{ 1,7 \right \}$ , making this the correct response.

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Question

Give the missing number in the sequence:

$12, 18, 24, 30, 36, 42, $\underline{; ; ; ; ; ; }$, 54...$

Tap to reveal answer

Answer

Each term in the sequence is generated from the previous term by adding 6,as follows:

$\textbf{12}+ 6 = \textbf{18 }$

$\textbf{18}+ 6 = \textbf{24 }$

$\textbf{24}+ 6 = \textbf{30 }$

$\textbf{30}+ 6 = \textbf{36 }$

$\textbf{36}+ 6 = \textbf{42 }$

$\textbf{42}+ 6 = \textbf{{\color{Red} 48}}$

The correct response is 48. This is confirmed by adding 6 again:

$\textbf{48}+ 6 = \textbf{54 }$

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Question

Give the missing number in the sequence:

$15, 19, 25, 33, 43, 55, $\underline{; ; ; ; ; ; }$, 85,...$

Tap to reveal answer

Answer

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:

$\textbf{15} + 4 = \textbf{19 }$

$\textbf{19} + 6 = \textbf{25 }$

$\textbf{25} + 8= \textbf{33 }$

$\textbf{33} + 10 = \textbf{43 }$

$\textbf{43} + 12 = \textbf{55 }$

$\textbf{55} + 14 = \textbf{{\color{Red} 69} }$ ,

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

$\textbf{69} + 16 = \textbf{85 }$

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Question

Give the missing number in the sequence:

$5, 6, 6, 8, 16, 19, 57, 61 , $\underline{; ; ; ; ; ; }$, 249, ...$

Tap to reveal answer

Answer

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

$\textbf{5}+ 1= \textbf{6}$

$\textbf{6} \times 1= \textbf{6}$

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

$\textbf{6}+ 2= \textbf{8}$

$\textbf{8} \times 2= \textbf{16}$

$\textbf{16}+ 3= \textbf{19}$

$\textbf{19} \times 3= \textbf{57 }$

$\textbf{57}+ 4= \textbf{61}$

$\textbf{61} \times 4= \textbf{{\color{Red} 244}}$ ,

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

$\textbf{244}+ 5 = \textbf{249}$ .

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Question

A car which gets 20 miles per gallon travels at a constant speed of 50 miles per hour. If the car's tank, which holds 16 gallons of gasoline, was filled at noon, when should the gas gauge read half full? (Assume no stops)

Tap to reveal answer

Answer

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

$$\frac{1}{2}$ \times 16 = 8$

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

$20 \textup{ mi/gal } \times 8 \textup{ gal} = 160 \textup{ mi}$ .

The car will travel at a constant speed of 50 miles per hour. We can set $r = 50 \textup{ mi /hr}$ and $d = 160 \textup{ mi }$ in the rate formula:

$t = $\frac{d}{r}$ = $\frac{160 \textup{ mi }$}{50 \textup{ mi /hr}} = 3 $\frac{1}{5}$\textup{ hr}$ .

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

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