Problem Solving - HSPT Math

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Question

Mr. and Mr. Smith have invited six guests to dinner. The above show their kitchen table with the locations of its eight seats.

How many ways can the eight participants be seated if the Smiths are to occupy seats 1 and 5?

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Answer

First, there are two ways to seat the Smiths; Mr. Smith at seat 1 and his wife at seat 5, and vice versa.

Second, the number of ways to seat the remaining six guests is the number of ways to permute 6 from 6:

$P (6, 6) = \frac{6!}{(6-6)!} = \frac{6!}{0!} = \frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}{1} = 720$

Multiply:

$2 \cdot 720 = 1,440$

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Question

Fill in the box with a single digit to form a number that is divisible by 24:

$13, 587, \square 26$ .

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Answer

A whole number divisible by 24 is also divisible by any factor of 24, including 8.

A whole number is divisible by 8 if and only if its last three digits form a number which itself is divisible by 8. We can try each of the three digits:

$126 \div 8 = 15 \textup{ R }6$

$426 \div 8 = 53 \textup{ R }2$

$726 \div 8 = 90 \textup{ R }6$

None of the choices yields a number divisible by 8. Since 8 cannot be a factor of this number, neither can 24.

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Question

Kevin is looking at the blueprint of the house he is building. The scale is 1 inch = 5 feet. On paper, the master bedroom is 2.5 inches by 3 inches. What is the actual size of the bedroom?

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Answer

The given dimensions must be multiplied by 5, so 2.5 inches becomes 12.5 feet, and 3 inches becomes 15 feet.

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Question

In state X, the fine for speeding is $75 plus $4 for each mile per hour over the limit that the person was driving. If the speed limit is and the driver's speed is , then which of the following formulas could be used to calculate the driver's fine?

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Answer

The number of miles per hour over the limit is the difference of the number of miles per hour the driver was going and the speed limit - this is . The fine is $75 plus $4 times this difference:

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Question

On a particular November day, it is $65^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-23^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $23^\circ F$ below zero and in Tucson it is $65^\circ F$ above zero; therefore we can say:

$23+65=88^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $88^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $70^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-27^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $27^\circ F$ below zero and in Tucson it is $70^\circ F$ above zero; therefore we can say:

$27+70=97^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $97^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $75^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-31^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $31^\circ F$ below zero and in Tucson it is $75^\circ F$ above zero; therefore we can say:

$31+75=106^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $106^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $80^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-33^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $33^\circ F$ below zero and in Tucson it is $80^\circ F$ above zero; therefore we can say:

$33+80=113^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $113^\circ F$ warmer than New Castle.

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Question

$\frac{\sqrt{25}*\sqrt{144}}{\sqrt{36}}=$

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Answer

First, solve the square roots:

$\sqrt{25}=5$

$\sqrt{144}=12$

$\sqrt{36}=6$

Then, solve accordingly:

$\frac{5*12}{6}=\frac{60}{6}=10$

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Question

On a particular November day, it is $85^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-32^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $32^\circ F$ below zero and in Tucson it is $85^\circ F$ above zero; therefore we can say:

$32+85=117^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $117^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $90^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-25^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $25^\circ F$ below zero and in Tucson it is $90^\circ F$ above zero; therefore we can say:

$25+90=115^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $115^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $95^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-26^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $26^\circ F$ below zero and in Tucson it is $95^\circ F$ above zero; therefore we can say:

$26+95=121^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $121^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $100^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-22^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $22^\circ F$ below zero and in Tucson it is $100^\circ F$ above zero; therefore we can say:

$22+100=122^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $122^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $105^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-20^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $20^\circ F$ below zero and in Tucson it is $105^\circ F$ above zero; therefore we can say:

$20+105=125^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $125^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $110^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-15^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $15^\circ F$ below zero and in Tucson it is $110^\circ F$ above zero; therefore we can say:

$15+110=125^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $125^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $115^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-10^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $10^\circ F$ below zero and in Tucson it is $115^\circ F$ above zero; therefore we can say:

$10+115=125^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $125^\circ F$ warmer than New Castle.

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Question

On a particular November day, it is $120^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-12^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

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Answer

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $12^\circ F$ below zero and in Tucson it is $120^\circ F$ above zero; therefore we can say:

$12+120=132^\circ F$

Also, we can solve this problem by using a number line. New Castle’s temperature is units away from zero and Tucson’s is units away.

We can see that Tucson is $132^\circ F$ warmer than New Castle.

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Question

You decide to buy a $500 table at a 20% mark down. You put a down payment of $40 then make twelve equal monthly payments. What is the amount of the payments?

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Answer

First you must find the selling price of the table since it was $500 but is now discounted 20%. Mathematically, a 20% discount is the same as the table being 80% the original price.

Also recall that a percentage can be written as a fraction or decimal.

$a%=\frac{a}{100}=0.0a$

Set up the following formula to solve.

$$500(1-0.20)= \textup{Selling Price}$500(0.80)=\textup{Selling Price}\ $500\times \frac{80}{100}=\textup{Selling Price}$

Multiply across and divide to find the selling price of the table.

Multiply across results in the following.

$$500\times 80=$40000$

Now divide by eighty.

$\frac{$40000}{100}=\frac{$400\times 100}{100}=\$400$

Now subtract the down payment from the selling price to get .

Finally you divide it by twelve.

$\frac{\$360}{12}=\frac{$30\cdot 12}{12}=$30$

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Question

What is the prime factorization of 48?

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Answer

We can find the prime factorization of a number by dividing 48 into smaller numbers.

$\dpi{100} 48=2\times 24$

2 is prime

$\dpi{100} 24=2\times 12$

2 is prime

$\dpi{100} 12=2\times 6$

2 is prime

$\dpi{100} 6=2\times 3$

Both 2 and 3 are prime.

So we have four 2's and one 3. Multiply these numbers to get 48.

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Question

How many whole numbers are between $\dpi{100} \frac{16}{5}$ and $\dpi{100} \frac{50}{7}$ ?

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Answer

Convert each fraction to a mixed number.

$\dpi{100} \frac{16}{5}=3\frac{1}{5}$

$\dpi{100} \frac{50}{7}=7\frac{1}{7}$

So how many whole numbers are between $\dpi{100} 3\frac{1}{5}$ and $\dpi{100} 7\frac{1}{7}$ ?

4, 5, 6, and 7 would fall between these two numbers. 3 would not. So there are 4 total whole numbers between them.

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