Quantitative Reasoning Questions

Seventh Grade

Let's get ready for some ERB questions in which you have to compare expressions. Remember, expressions are just different ways of expressing possible values. They usually contain some combination of numbers and letters.

1. What's greater: 3a or 2a?

Remember that 3a means 3 times a. The letter a is a variable. It's job in this expression is to stand in for an unknown number. When you have a number in front of a letter, it means you need to multiply the number by the unknown. So, 3a means 3 times an unknown number, which we are calling a.

First, do you think that we know enough about the unknown to answer this question? Try substituting some actual numbers for a.
a. If a is any positive integer, what's greater: 3a or 2a?
b. But if a is zero, what's greater: 3a or 2a?
c. If a is any negative integer, what's greater: 3a or 2a?

So, you should have gotten 3 different answers, depending on what you substituted for a in those 3 different examples above. If not, think again!!

Okay, adding on to that:

2. What's greater: 2(a + 2) or 2a + 3?

First, we have to use the distributive property to multiply 2(a + 2). In other words, we distribute the multiplication to each of the two terms in the parenthesis. So,
2(a + 2) = (2 x a) + (2 x 2) = 2a + 4

Now, let's ask again the same question, now using the restated version of 2(2 + 2)

What's greater: 2a + 4 or 2a + 3?

It is easier to see now that we are comparing apples to apples. The question becomes simpler. Now, does it matter that we don't know whether the unknown a is:
a. a positive integer
b. zero
c. a negative integer?

You should see that the left side of this equation is always going to be greater, because the 2a part of the expression will be equal. So, we are able to say that 4 is always greater than 3!!! This is true no matter what value we substitute for a!!

3. Now let's ask, what's greater if a is an integer: a or - a?

Isn't this a lot like the first question?

4. What is the LCD (lowest common denominator) for 5/6 and 1/9?
Hint: the LCD is the same as the LCM for the two denominators.

Finding the LCM:
multiples of 6: 6, 12, 18, 24 30
multiples of 9: 9, 18, 27, 36

The lowest multiple they have in common is 18! So, the lowest common denominator for 6ths and 9ths will be 18.

We need LCD to add and subtract fractions, although we can use any common denominator that works. So, let's add those two fractions:

5/6 gets restated as 15/18.
1/9 gets restated as 2/18.

Once you have the common denominators, just add the numerators and leave the denominators alone!!

Now, the ERB has a lot of questions on Prime Numbers, so let's do some prime number practice: primes are numbers greater than one with only two factors: themselves and one. List begins: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47........

Now, try not to look at the list and do these problems:
1. What is the number equal to the sum of the first prime number after 20 plus the greatest prime number less than 5?

2. What is the number equal to the sum of the first prime after 30 plus the greatest prime number less than 10?

Translating English to expressions and back again:
1. A and B each brought x marbles to school. C brought 3 more marbles than either of them. Altogether how many marbles were brought to school?

x + x + x + 3 simplifies to 3x + 3
That's an expression that tells you how many marbles they all brought.

2. Try this one yourself: A and B each brought y marbles to school and C brought 200 more than either of them. Altogether, how many marbles were brought to school?


3. Now, let's use multiplication in our expression, remembering what we did at the very beginning for expressing multiplication with a variable.

Joe ate n sandwiches and Jack ate three times as many as Joe. Altogether, how many sandwiches did the two boys eat?

n + 3n which then simplifies to 4n

4. Try this one yourself: Bill ate n grapes, Bob ate twice as many as Bill, and Jack ate 5 times as many as Bill. Altogether, how many grapes did they eat?


5. Now, what if we say, Bill ate n, Bob ate twice what Bill did, and Jack ate 5 times what Bob did. That expression would look different, like this:
n + 2n + 5(2n) which can simplify to n + 2n + 10n and further to 13n

So, 13n would be the expression which tells you how many they ate altogether.