ISEE Middle Level Quantitative Reasoning - Numbers and Operations

Figures 1

Refer to the above diagrams. Each figure is divided into sections of equal size and shape.

Which is the greater quantity?

(a) The fraction of Figure 1 that is shaded

(b) The fraction of Figure 2 that is shaded

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(b) is the greater quantity

Explanation

Figure 1 is a rectangle divided into 24 squares of equal size; 8 of the squares are shaded, which means that $\frac{8}{24}$ of Figure 1 is shaded.

Figure 2 is a triangle divided into 4 triangles of equal size; 1 is shaded, which means that $\frac{1}{4} = \frac{1 \times 6 }{4 \times 6} =\frac{6}{24 }$ of Figure 2 is shaded.

The fraction of Figure 1 that is shaded is the greater quantity.

is a positive number. Which is the greater quantity?

(a) The reciprocal of $Y - \frac{1}{2}$

(b) The reciprocal of $Y + \frac{1}{2}$

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation

We examine two scenarios to demonstrate that the given information is insufficient.

Case 1: $Y = 2\frac{1}{2}$

$Y - \frac{1}{2} = 2 \frac{1}{2 }- \frac{1}{2} = 2$

The reciprocal of this is $\frac{1}{2}$ .

$Y + \frac{1}{2} = 2 \frac{1}{2 }+ \frac{1}{2} = 3$

The reciprocal of this is $\frac{1}{3}$ .

$\frac{1}{2} > \frac{1}{3}$ , so $Y - \frac{1}{2}$ has the greater reciprocal.

Case 2: $Y = \frac{1}{4}$ .

$Y - \frac{1}{2} = \frac{1}{4 }- \frac{1}{2} = \frac{1}{4 }- \frac{2}{4} = - \frac{1}{4 }$

The reciprocal of this is .

$Y + \frac{1}{2} = \frac{1}{4 }+ \frac{1}{2} = \frac{1}{4 }+ \frac{2}{4} = \frac{3}{4 }$

The reciprocal of this is $\frac{4 }{3}$ .

$\frac{4 }{3} > -4$ , so $Y + \frac{1}{2}$ has the greater reciprocal.

Untitled

The square and the triangle in the above diagram are both equally divided. Which is the greater quantity?

(a) The fraction of the square that is shaded

(b) The fraction of the triangle that is shaded

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

It is impossible to determine which is greater from the information given

Explanation

The square is divided into eighteen triangles of equal size and shape; nine are shaded, so the fraction of the square that is shaded is

$\frac{9}{18} = \frac{9 \div 9 }{18\div 9 } = \frac{1}{2}$

The triangle is divided into sixteen triangles of equal size and shape; eight are shaded, so the fraction of the square that is shaded is

$\frac{8}{16} = \frac{8 \div 8 }{16 \div 8 } = \frac{1}{2}$ .

The fractions are equal.

Solve the following:

$\frac{3}{4} \cdot \frac{1}{2}$

$\frac{3}{8}$

$\frac{3}{2}$

$\frac{1.5}{4}$

$\frac{1.5}{2}$

$\frac{1}{2}$

Explanation

To multiply equations, we will multiply the numerators together, then we will multiply the denominators together. So, we get

$\frac{3}{4} \cdot \frac{1}{2}$

$\frac{3 \cdot 1}{4 \cdot 2}$

$\frac{3}{8}$

Danny drives 70 miles per hour. How long will it take him to reach his grandfather's house 245 miles away.

Explanation

Divide $245 \div 70=3.5$ .

Answer: It will take Danny 3.5 hours to arrive to his grandfather's house.

One Mexican peso is equal to about 7.4 cents. To the nearest whole number, for how many pesos can a tourist to Mexico expect to exchange $500?

Explanation

One peso is equivalent to 7.4 cents, or $0.074, so divide $500 dollars by this conversion factor. $500 is equivalent to

$500 \div 0.074 \approx 6,757$ pesos.

What is the decimal equivalent of $\frac{36}{90}$ ?

Explanation

Divide.

$36\div 90=0.40$

Another way to solve is to simplify the fraction by removing common factors.

$\frac{36}{90}=\frac{36\div9}{90\div9}=\frac{4}{10}$

Four over ten, or four-tenths, is equal to

Answer:

One euro is worth approximately $1.27. For how many euros (nearest whole) can an American tourist expect to exchange $1,000?

Explanation

One Euro is equivalent to $1.27, so divide the number of dollars by this conversion factor. $1,000 is equivalent to

$1,000 \div 1.27\approx 787$ euros

Multiply the following:

$\frac{2}{3} \cdot 18$

Explanation

To multiply fractions, we will multiply the numerators together, then we will multiply the denominators together. Note that we do NOT need to find a common denominator.

So, in the problem

$\frac{2}{3} \cdot 18$

We will first write 18 as a fraction. We know that whole numbers can be written as a fraction over 1. So, we get

$\frac{2}{3} \cdot \frac{18}{1}$

Now, before we multiply, we can simplify to make things easier. The 3 and the 18 can both be divided by 3. We get

$\frac{2}{1} \cdot \frac{6}{1}$

Now, we can multiply straight across. We get

$\frac{2 \cdot 6}{1 \cdot 1}$

$\frac{12}{1}$

$\tfrac{3}{4} \times 12 =$

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{1}{12}$

Explanation

When dealing with problems that involve fractions and whole numbers, first convert your whole number into a fraction. You can easily do this by putting the whole number over 1.

$12 \rightarrow\tfrac{12}{1}$