Number Concepts and Operations - SSAT Upper Level Quantitative (Math)

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Question

$\dpi{100} .24+2.5+\frac{1}{4}=$

Answer

First, convert $\dpi{100} \frac{1}{4}$ into a decimal. We are left with

$\dpi{100} .24+2.5+.25$

Now add the three numbers, being careful to line the decimal points up exactly. From right to left we will add:

4 and 5 = 9

5, 2, and 2 = 9

2 is alone = 2

This leaves us with 2.99 for the final answer.

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Question

$Solve: 7+(3-9)^{2}+36\div 4$

Answer

Use order of operations, so you would first complete the operation in the parentheses, which would give you .

Then you do exponenets, so $\left ( -6\right)^{2} = 36$ .

Then you do division which is $36\div4 = 9$ .

Then you do addition from left to right, so .

Last, you add to , which is .

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Question

What is the sum of the two percentages in decimal form?

7 percent and 135 percent

Answer

You add the percentages and you get 142 percent. To convert a percent to a decimal, you divide by 100, so the answer is 1.42.

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Question

$3+8\div4-(2\times 5)^{^{2}}$

Answer

Do the expression in the parantheses and you get . Then you do $10^{2}$ and you get . Then you do $8\div4$ and you get . Then you add .

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Question

If Susie bought a pair of shoes for , a shirt for , and a pair of pants for . How much did she spend in total?

Answer

You add all three values and you get .

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Question

and are prime integers. and . What is the greatest possible value of ?

Answer

To find the greatest possible value of , add the greatest possible value of to the greatest possible value of . This happens when is equal to the greatest prime between 35 and 45, which is 43, and when is equal to the greatest prime between 85 and 95, which is 89. Add the two numbers:

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Question

and are prime integers: and . How many possible values of are there?

Answer

can be equal to any of the prime numbers between 45 and 55, of which there are two - 47 or 53.

can be equal to any of the prime numbers between 15 and 25, of which there are three - 17, 19, and 23.

Their sum can be any of the following:

This makes five possible sums.

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Question

In a math contest, the team from Lincoln High comprised five students - Melanie, Ned, Olivia, Phyllis, and Quinn. Melanie outscored Ned by 12 points; Ned outscored Quinn by 19 points; Phyllis outscored Ned by 45 points and Olivia by 20 points. Who came in third among the five?

Answer

Since this reasoning is valid regardless of the actual scores, suppose Quinn scored 100. Then Ned scored 19 more than Quinn, or 119, and Melanie scored 12 more than Ned, or 131. Phyllis scored 45 more than Ned, or 164, and Olivia scored 20 points fewer than Phyllis, or 144. This makes the order, from greatest to least:

Phyllis, Olivia, Melanie, Ned, Quinn

Melanie scored third among the five.

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Question

Which of the following angles is supplementary to an angle that is 45 degrees?

Answer

Supplementary angles add up to 180 degrees. Given that , the correct answer is 135.

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Question

Simplify:

Answer

To simplify this expression, regroup and combine like-terms.

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Question

Add:

$\frac{\frac{1}{5}+\frac{5}{6}}{\frac{1}{4}+\frac{2}{5}}+\frac{5}{6}=?$

Answer

First, add the fractions found in the numerator and denominator of the complex fraction. To add two fractions, they need to have a common denominator. You can create a common denominator by multiplying the numerator and denominator of one fraction by as a fraction made up of the denominator of the other fraction. For example, if you need to create a common denominator for $\frac{1}{3}$ and $\frac{1}{2}$ , you would multiply $\frac{1}{3}$ by $\frac{2}{2}$ and $\frac{1}{2}$ by $\frac{3}{3}$ . Since this just multiplies each fraction by , it doesn't change the fractions' values, just the way in which they are represented.

$\frac{1}{5}+\frac{5}{6}=(\frac{1}{5}\cdot \frac{6}{6})+(\frac{5}{6}\cdot\frac{5}{5})=\frac{6}{30}+\frac{25}{30}=\frac{31}{30}$

$\frac{1}{4}+\frac{2}{5}=(\frac{1}{4}\cdot\frac{5}{5})+(\frac{2}{5}\cdot\frac{4}{4})=\frac{5}{20}+\frac{8}{20}=\frac{13}{20}$

Now, divide those two fractions.

$\frac{31}{30}\div\frac{13}{20}=\frac{31}{30}\times\frac{20}{13}=\frac{62}{39}$

Finally, add $\frac{5}{6}$ :

$\frac{62}{39}+\frac{5}{6}=(\frac{62}{39}\cdot \frac{6}{6})+(\frac{5}{6}\cdot\frac{39}{39})=\frac{372}{234}+\frac{195}{234}=\frac{567}{234}\div 3=\frac{189}{78}\div 3=\frac{63}{26}$

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Question

Solve.

$\frac{6}{7x}+\frac{3}{2x^{2}}$

Answer

Find the least common denominator between the two fractions. In our case both and go into $14x^{2}$ thus, this is our common denominator.

The top is multiplied by 2x for the left fraction and to get:

$14x^{2} \rightarrow \frac{6}{7x}\cdot \frac{2x}{2x}=\frac{12x}{14x^2}$ .

For the right fraction, the top is multiplied by 7.

$\frac{3}{2x^2}\cdot \frac{7}{7}=\frac{21}{14x^2}$

To solve we need to add these two fractions together.

Final answer should read,

$\frac{12x}{14x^2}+\frac{21}{14x^2}=\frac{12x+21}{14x^2}$ .

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Question

Solve.

$\frac{6}{x^{2}+6x+9}+\frac{x-2}{x+3}$

Answer

By inspection, the denominator of the left fraction is a perfect square $\left ( x+3\right )^2$ . Since the right fraction already has a , just multiply top and bottom by so the denominators of both fraction are the same.

$\frac{6}{x^2+6x+9}+\frac{(x-2)(x+3)}{(x+3)(x+3)}$

$\frac{6}{x^2+6x+9}+\frac{x^2+3x-2x-6}{x^2+6x+9}$

The top should read with a bottom of $\left ( x+3\right )^2$ .

After simplification, the answer is revealed.

$\frac{6+x^2+x-6}{x^2+6x+9}=\frac{x^2+x}{x^2+6x+9}$

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Question

Simplify.

$\frac{1-x}{x-1}+\frac{x-1}{1-x}$

Answer

If you can recognize that $\frac{1-x}{x-1}$ is -1 then the answer is quick.

Otherwise just find the least common denominator which is .

The numerator of the new fraction will be . It may be tempting to think $\frac{2x^{2}-4x+2}{-x^2+2x-1}$ should have been the right answer, however, the question does say simplify and yes this can be simplified. If you factor the out of the numerator, it's . The bottom does look very similar to the top only the signs are flipped. This means if you factor out a on the bottom, the quadratic equation should match and cancel out leaving you with the answer of .

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Question

Solve.

$\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{5}+\frac{1}{8}}+\frac{1}{3}$

Answer

First focus on the big fraction specificall,y the numerator. We need to find a common denominator in the numerator in order to add the two fractions and simplify the expression.

The top should have a common denominator of .

$\frac{1}{2}\cdot \frac{7}{7}+\frac{1}{7}\cdot \frac{2}{2}=\frac{7}{14}+\frac{2}{14}$

Therefore the top should become $\frac{9}{14}$ .

Now lets look at the denominator of the big fraction. We again need to find a common denominator for the two components. In this case it will be .

$\frac{1}{5}\cdot \frac{8}{8}+\frac{1}{8}\cdot \frac{5}{5}=\frac{8}{40}+\frac{5}{40}$

Therefore the bottom should become $\frac{13}{40}$ .

Since we need to have a single fraction added to a $\frac{1}{3}$ , we need to multiply top and bottom by $\frac{40}{13}$ so we can add the fractions easier.

$\frac{\frac{9}{14}}{\frac{13}{40}}=\frac{9}{14}\cdot \frac{40}{13}=\frac{360}{182}=\frac{180}{91}$

So far, the new equation to solve is $\frac{180}{91}$ $+\frac{1}{3}$ .

Same thing, find the least common denominator and solve to arrive at the final answer.

$\frac{180}{91}\cdot \frac{3}{3}+\frac{1}{3}\cdot \frac{91}{91}=\frac{540}{273}+\frac{91}{273}=\frac{631}{273}$

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Question

Solve.

$1+\frac{1}{1+\frac{1}{x}}$

Answer

Although intimidating, just focus on the inner fraction and work your way up.

Common denominator is so that should lead to

$1+\frac{1}{1\cdot \left(\frac{x}{x} \right )+\frac{1}{x}} \rightarrow$ $1+\frac{1}{\frac{x+1}{x}}$ .

By multiplying everything by $\frac{x}{x+1}$ this should lead to $1+\frac{x}{x+1}$ .

Common denominator is now and this should arrive at,

$1\cdot \frac{x+1}{x+1}+\frac{x}{x+1}=\frac{x+1+x}{x+1}=\frac{2x+1}{x+1}$

which is the final answer when everything is simplified.

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Question

Simplify.

$\frac{1}{1+\frac{1}{1+x}}+\frac{2}{2+\frac{2}{2+x}}$

Answer

Work on each fraction and start from the bottom and work your way up.

For the left fraction, it should be

$\frac{1}{\frac{x+2}{x+1}}$ since is the common denominator.

Then multiply by the inverse to get $\frac{x+1}{x+2}$ .

For the right, it's the same concept as the final fraction to get is $\frac{x+2}{x+3}$ .

Finally find the common denominator of these new complex fractions and solve to get the final answer.

$\frac{x+1}{x+2}\cdot \frac{x+3}{x+3}+\frac{x+2}{x+3}\cdot \frac{x+2}{x+2}$

$\frac{x^2+4x+3}{x^2+5x+6}+\frac{x^2+4x+4}{x^2+5x+6}$

$\frac{2x^2+8x+7}{x^2+5x+6}$

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Question

Simplify.

$\frac{1}{x}+\frac{2}{2x}+\frac{3}{3x}+\frac{4}{4x}+\frac{5}{5x}$

Answer

By careful inspection, the fractions all reduce to $\frac{1}{x}$ .

$\frac{2}{2x}=\frac{2\cdot 1}{2\cdot x}=\frac{1}{x}$

$\frac{3}{3x}=\frac{3\cdot 1}{3\cdot x}=\frac{1}{x}$

$\frac{4}{4x}=\frac{4\cdot 1}{4\cdot x}=\frac{1}{x}$

$\frac{5}{5x}=\frac{5\cdot 1}{5\cdot x}=\frac{1}{x}$

All the fractions have values in the numerator and denominator that cancel out.

There are five of these so multiply $\frac{1}{x}$ by to get the final answer.

Answer should be $\frac{5}{x}$ .

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Question

Simplify.

$\frac{x^2+5x+6}{x^2+9x+14}+\frac{x^2+6x+5}{x^2+7x+6}$

Answer

Don't try to find the least common denominator as it will take a lot of time and a definite mistake in arithmetic. Instead try to see if these quadratics can simplify.

Upon inspection, we get

$\frac{(x+2)(x+3)}{(x+2)(x+7)}+\frac{(x+1)(x+5)}{(x+1)(x+6)}$ .

Remember, to break down the quadratic equation by finding the binomials, two numbers that are factors of c must add up to make b.

With the cancellations, we should get

$\frac{x+3}{x+7}+\frac{x+5}{x+6}$ .

The common denominator will be

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables. The left fraction numerator is multiplied by and the right fraction numerator is multiplied by .

Overall, the new fraction should read

$\frac{x^2+6x+3x+18+x^2+7x+5x+35}{x^2+13x+42}$ .

After simplification, answer should be $\frac{2x^2+21x+53}{x^2+13x+42}$ .

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Question

Simplify.

$\frac{2x^2+2x-4}{x^2+2x-3}+\frac{x^2-3x-10}{2x^2-8x-10}$

Answer

Try to simplify the fraction rather than finding the least common denominator. By breaking the quadratic equation into its factors, it becomes:

$\frac{2(x+2)(x-1)}{(x-1)(x+3)}+\frac{(x-5)(x+2)}{2(x-5)(x+1)}$ .

Remember, to break down the quadratic equation by finding the binomials, two numbers that are factors of c must add up to make b. Also, if the values of a, b, and c in the quadratic equation can be reduced, then factor out that divisor to make the quadratic easier to factor.

Upon cancelling, we should get

$\frac{2(x+2)}{(x+3)}+\frac{(x+2)}{2(x+1)}$ .

The common denominator is $(x+3)\left ( 2x+2\right )$ or or . Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables. Then multiply the numerator of left fraction by and the numerator of right fraction by and the new fraction should look like,

$\frac{4(x^2+x+2x+2)+x^2+2x+3x+6}{2x^2+8x+6}$ .

Upon distributing the four and simplifying the fraction, answer is shown.

$\frac{5x^2+17x+14}{2x^2+8x+6}$

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