SSAT Upper Level Quantitative - How to solve algebraic word problems

Michael scores a 95, 87, 85, 93, and a 94 on his first 5 math tests. If he wants a 90 average, what must he score on the final math test?

Explanation

To solve for the final score:

Add the five past test scores and you get 454. Then set up an algebraic equation where you add 454 to , which is the final test score, and divide by six, because you want the average for 6 tests now. You make this equation equal to 90 because that is the average Michael wants and solve for :

$\left ( 454+x\right )\div6 = 90$

Beth and Sam are 500 miles apart. If Beth travels at 60mph and leaves her house at 1pm, what time will she arrive at Sam's house?

9:20 PM

9:00 PM

9:33 PM

9:30 PM

8:33 PM

Explanation

Using $Distance = rate\times time$ , the time would be hours, which is hours and minutes. If you add that to 1pm, you get 9:20pm.

Eddie, Freida, Grant, Helene, and Ira represented Washington High in a math contest. The team score was the sum of the three highest scores. Grant outscored Eddie and Freida; Helene outscored Grant; Freida outscored Ira. Which three students' scores were added to determine the team score?

Insufficient information is given to answer the question.

Freida, Grant, and Helene

Eddie, Grant, and Helene

Grant, Helene, and Ira

Eddie, Grant, and Ira

Explanation

Let be Eddie's, Freida's, Grant's, Helene's and Ira's scores. Each of the following statements can be translated into inequalities as follows:

Grant outscored Eddie and Freida:

Helene outscored Grant:

Freida ourtscored Ira:

The first and third statements can be combined to form the three-part inequality:

The second, third, and fourth statements can be combined to form the four-part inequality:

Since Helene and Grant were the top two finishers, their scores were counted. However, it cannot be determined which student finished third from these statements. Therefore, insufficient information is given to answer the question.

The price of a nugget is units, which is two more than twice the value of a bronze object. What is the price of the bronze object?

$\frac{3}{2}$

$\frac{2}{3}$

$\frac{5}{2}$

$\frac{2}{5}$

Explanation

Write the word problem in terms of a mathematical equation. Let be the value of the bronze object.

Solve for .

$x=\frac{3}{2}$

The value of the bronze object is $\frac{3}{2}$ units.

A pitcher standing on top of a 96-foot high building throws a baseball straight up at an initial speed of 80 miles per hour. The height in feet of the ball after time seconds can be modelled by the equation

$h(t)= -16t^{2}+ 80t+ 96$ .

How long does it take for the ball to hit the ground?

6 seconds

4 seconds

8 seconds

10 seconds

12 seconds

Explanation

When the ball hits the ground, the height is 0, so set and solve for :

$-16t^{2}+ 80t+ 96 = 0$

$-16(t^{2}- 5t-6) = 0$

Either or .

If , then . Since time cannot be negative, we throw this out.

If , then - this is the answer we accept.

The ball hits the ground in 6 seconds.

Which of the following sentences is represented by the equation

$\sqrt{x + 5} = x + 6$ ?

The square root of the sum of a number and five is six greater than the number.

The square root of the sum of a number and five is six less than the number.

The sum of five and the square root of a number is six greater than the number.

The sum of five and the square root of a number is six less than the number.

None of the other responses are correct.

Explanation

$\sqrt{x + 5}$ is the square root of , which in turn can be written as "the sum of a number and five"; $\sqrt{x + 5}$ can subsequently be written as "the square root of the sum of a number and five". Since $\sqrt{x + 5} = x + 6$ , $\sqrt{x + 5}$ is six greater than , the number, so the equation can be stated as "The square root of the sum of a number and five is six greater than the number."

What is the mean of the set below?

$\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{1}{12}$

$\frac{7}{24}$

$\frac{1}{2}$

Explanation

The first step is to convert the set $\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{1}{12}$ to fractions that have a common denominator of 12. This gives us:

$\frac{3}{12}, \frac{4}{12}, \frac{6}{12}, \frac{1}{12}$

The mean is then calculated by dividing the sum of the numbers in the set by the number of items in the set.

The sum of the items in the set is: $\frac{3}{12}+\frac{4}{12}+\frac{6}{12}+ \frac{1}{12}=\frac{14}{12}$

There are 4 items in the set, so the sum must be divided by 4 (or multiplied by $\frac{1}{4}$ ).

This results in:

$\frac{14}{12}\cdot\frac{1}{4}=\frac{14}{48}=\frac{7}{24}$

The mean of , , , and is 125; the mean of , , , and is 150. Which of the following gives the sum of and if the mean of , , , , , and is ?

Explanation

Since the mean of the four numbers , , , and is 125,

$\frac{a + b + c + d}{4} = 125$

$\frac{a + b + c + d}{4} \cdot 4 = 125\cdot 4$

Similarly,

$\frac{ c + d+e+f}{4} = 150$

$\frac{ c + d+e+f}{4}\cdot 4 = 150\cdot 4$

Add the two sums:

The mean of , , , , , and is , so

$\frac{a + b + c + d + e + f }{6}= M$

$\frac{a + b + c + d + e + f }{6}\cdot 6= M \cdot 6$

So:

If David wants to drive to his friend's house, which is 450 miles away, in 6 hours, what is the average speed David has to drive at?

$75\ mph$

$50\ mph$

$60\ mph$

$80\ mph$

$65\ mph$

Explanation

$Distance=rate\times time$

Plug in the the values for distance and time, and solve for rate.

$450 = r\times 6$

and

Greg is trying to fill a 16 oz. bottle with water. If Greg fills the bottle at 1 oz per second and the bottle leaks .2 oz per second, how long would it take for Greg to fill the bottle?

$20\ seconds$