HSPT Math : How to do other word problems

Study concepts, example questions & explanations for HSPT Math

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Example Questions

Example Question #504 : Problem Solving

A Special Pizza at Monorail Deli has four different toppings; the customer can choose any one meat from pepperoni, sausage, hamburger, and anchovy, and any two vegetables from mushrooms, onions, black olives, green olives, tomatoes, and green peppers. The fourth topping can be a meat or a vegetable.

Clara wants to order a Special Pizza; however, she does not like pepperoni, and she is allergic to onions. How many possible ways can four toppings be chosen for a Special Pizza in order to meet Clara's specifications?

Possible Answers:

\displaystyle 60

\displaystyle 30

\displaystyle 45

\displaystyle 90

Correct answer:

\displaystyle 60

Explanation:

Clara has three meats from which to choose (the four available, minus the one she will not eat); she has five vegetables from which to choose (the six available minus the one to which she is allergic). She can do one of the following:

Case 1: She can choose one meat and three vegetables.

She can choose one of three meats. 

She can choose three of five vegetables; the number of ways to do so is

\displaystyle C(5,3)

\displaystyle = \frac{5!}{(5-3)! 3!}

\displaystyle = \frac{5!}{2! 3!}

\displaystyle = \frac{1 \cdot 2 \cdot 3 \cdot 4\cdot 5 }{1 \cdot 2 \cdot 1 \cdot 2\cdot 3 }

\displaystyle = \frac{ 4 \cdot 5 }{ 1 \cdot 2 }

\displaystyle = \frac{20}{2}

\displaystyle = 10

The number of possible pizzas with one meat and three vegetables is 

\displaystyle 3 \cdot C(5,3) = 3 \cdot 10 = 30

Case 2: She can choose two meats and two vegetables.

She can choose two of three meats. There are three ways to do this, since choosing two meats means leaving one out; there are three ways to choose which meat to leave out.

She can choose two of five vegetables; the number of ways to do so is

\displaystyle C(5,2)

\displaystyle = \frac{5!}{(5-2)! 2!}

\displaystyle = \frac{5!}{3! 2!}

\displaystyle = \frac{1 \cdot 2 \cdot 3 \cdot 4\cdot 5 }{1 \cdot 2 \cdot 3 \cdot 1 \cdot 2 }

\displaystyle = \frac{ 4 \cdot 5 }{ 1 \cdot 2 }

\displaystyle = \frac{20}{2}

\displaystyle = 10

The number of possible pizzas with two meats and two vegetables is 

\displaystyle 3 \cdot C(5,2) = 3 \cdot 10 = 30

The total number of combinations of toppings from which Clara can choose is 

\displaystyle 30+30 = 60

 

Example Question #31 : How To Do Other Word Problems

Rent in Melissa's apartment is $575 per month. The rent is due on the fifth of the month, and Melissa must pay $25 penalty per day late. 

Melissa moved into the apartment on February 15 and agreed to pay a prorated rent of $250 for February. She also paid a $300 security deposit. For the remainder of the year, she paid the regular rent on the first of each month, except for October, when she paid on the fourth of the month. Calculate the amount of money she paid the apartment complex in 2014, including the security deposit and any penalties.

Possible Answers:

\displaystyle \$6, 375

\displaystyle \$6, 300

\displaystyle \$6, 950

\displaystyle \$6, 875

Correct answer:

\displaystyle \$6, 300

Explanation:

Melissa paid $575 rent for each month from March to December - ten months - for a total of 

\displaystyle \$ 575 \times 10 = \$5, 750.

Add to this her prorated February rent and her security deposit (she was not late with any rent payments, hence no penalties); Melissa paid

\displaystyle \$( 5, 750 + 250 +300)= \$6, 300

Example Question #62 : Word Problems

Rent in Henry's apartment is $645 per month. The rent is due on the fifth of the month, and Henry must pay $20 penalty per day late.

In 2014, Henry paid the rent on time each month except in November, when he paid the rent on November 8. How much rent did Henry pay for the year, including the penalty for November?

Possible Answers:

\displaystyle \$7,800

\displaystyle \$7,740

\displaystyle \$7,760

\displaystyle \$7, 680

Correct answer:

\displaystyle \$7,800

Explanation:

Henry's rent is $645 per month; multiply this by 12 to get yearly rent before the penalty.

\displaystyle \$ 645 \times 12 =\$ 7,740

Henry paid his November rent three days late, so multiply the daily penalty of $20 by 3 to get the total penalty.

\displaystyle \$20 \times 3 = \$ 60

Add the regular yearly rent and the November penalty:

\displaystyle \$ 7,740 + \$60 = \$7, 800

 

Example Question #31 : How To Do Other Word Problems

Consider the sequence 

What number replaces the square?

Possible Answers:

\displaystyle 102

\displaystyle 134

\displaystyle 94

\displaystyle 64

Correct answer:

\displaystyle 134

Explanation:

The increment that is added to each successive term to obtain the next term is doubled each time.

\displaystyle 7 + 1 = 8

\displaystyle 8 + 2 = 10

\displaystyle 10 + 4 = 14

\displaystyle 14+ 8 = 22

\displaystyle 22+ 16 = 38

\displaystyle 38 + 32 = 70, the number in the circle

\displaystyle 70 + 64 = 134, the number in the square, which is the correct choice.

Example Question #71 : Word Problems

Consider the sequence 

What number replaces the square?

Possible Answers:

\displaystyle 94

\displaystyle 76

\displaystyle 65

\displaystyle 47

Correct answer:

\displaystyle 76

Explanation:

Beginning with the third term, each term is the sum of the previous two:

\displaystyle 1+3 = 4

\displaystyle 3+4 = 7

\displaystyle 4+ 7 = 11

\displaystyle 7+11 = 18

\displaystyle 11+18 = 29

\displaystyle 18 + 29 = 47, the number in the circle;

\displaystyle 2 9+ 47 = 76, the number in the square.

Example Question #512 : Hspt Mathematics

Consider the sequence

\displaystyle \bigcirc, 4, 8 , 14, 22, 32, 44,...

What number replaces the circle?

Possible Answers:

\displaystyle 4

\displaystyle 0

\displaystyle 3

\displaystyle 2

Correct answer:

\displaystyle 2

Explanation:

The number that is added to each term to obtain the next one increases by 2 each time:

\displaystyle 4+ 4 = 8

\displaystyle 8+6 = 14

\displaystyle 14 + 8 = 22

\displaystyle 22+10 = 32

\displaystyle 32 + 12 = 44

If we let \displaystyle X be the number in the circle, then to maintain the pattern,

\displaystyle X + 2 = 4

so

\displaystyle X = 2. This is the correct choice. 

Example Question #513 : Hspt Mathematics

Consider the sequence

\displaystyle \bigcirc, 4, 11, 15, 26, 41, 67, 108,...

What number replaces the circle?

Possible Answers:

\displaystyle 1

\displaystyle 5

\displaystyle 7

\displaystyle 3

Correct answer:

\displaystyle 7

Explanation:

Each term, beginning with the third, is the sum of the previous two:

\displaystyle 4+ 11 = 15

\displaystyle 11+ 15 = 26

\displaystyle 15+26 = 41

\displaystyle 26 + 41 = 67

\displaystyle 41+ 67 = 108

If \displaystyle X is the term in the circle, then 

\displaystyle X+ 4 = 11

and

\displaystyle X = 7.

Example Question #31 : How To Do Other Word Problems

An arithmetic sequence begins

What number replaces the square?

Possible Answers:

\displaystyle 22

\displaystyle 19

\displaystyle 20

\displaystyle 21

Correct answer:

\displaystyle 19

Explanation:

In an arithmetic sequence, each term is obtained by adding the same number to the previous term - the common difference. Since \displaystyle 7 - 1 = 6, 6 is the common difference. The next two terms are:

\displaystyle 7+6 = 13

\displaystyle 13+6 = 19

19 replaces the square and is the correct choice.

Example Question #515 : Hspt Mathematics

A geometric sequence begins

What number replaces the square? Give the closest answer.

Possible Answers:

\displaystyle 250

\displaystyle 350

\displaystyle 400

\displaystyle 300

Correct answer:

\displaystyle 350

Explanation:

In a geometric sequence, each term is obtained by muliplying the previous term by the same number - the common ratio. 

Since \displaystyle 7 \div 1 = 7, 7 is the common ratio. The next two numbers in the sequence are 

\displaystyle 7 \times 7 = 49

\displaystyle 49 \times 7 = 343

343 replaces the square; of the four choices, 350 comes closest and is the correct response.

Example Question #513 : Problem Solving

A Special Pizza at Deli Llama has four different toppings; the customer can choose any one meat from pepperoni, sausage, hamburger, and anchovy, and any three vegetables from mushrooms, onions, black olives, green olives, tomatoes, and green peppers.

Mike and John want to order a Special Pizza. Mike doesn't like anchovies, and John is allergic to mushrooms. How many ways can the toppings of a Special Pizza be chosen to both men's specifications?

Possible Answers:

\displaystyle 30

\displaystyle 40

\displaystyle 60

\displaystyle 80

Correct answer:

\displaystyle 30

Explanation:

There are three ways to choose a meat topping - four minus the one Mike won't eat.

There are five vegetable toppings - six minus the one to which John is allergic - from which to choose three. This is

\displaystyle C(5,3)

\displaystyle = \frac{5!}{(5-3)! 3!}

\displaystyle = \frac{5!}{2! 3!}

\displaystyle = \frac{1 \cdot 2 \cdot 3 \cdot 4\cdot 5 }{1 \cdot 2 \cdot 1 \cdot 2\cdot 3 }

\displaystyle = \frac{ 4 \cdot 5 }{ 1 \cdot 2 }

\displaystyle = \frac{20}{2}

\displaystyle = 10

The number of ways to make a Special Pizza that leaves out anchovies and mushrooms is 

\displaystyle 3 \cdot C(5,3) = 3 \cdot 10 = 30

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