Mathematics: Quantitative Reasoning Study Guide for the TSIA2

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Decimals

Understanding decimals is based on understanding place value. Here is a table showing the place values for \(0.5124\). (Five thousand one hundred twenty-four ten-thousandths) You might think that the zero is not necessary, but it’s good practice to include it to keep from overlooking the decimal point.

\[\begin{array}{|c|c|c|c|c|c|} \hline 0 & . &5&1&2&4\\ \hline \times\; 1& &\times\; 0.1 & \times\; 0.01 & \times \;0.001 & \times \;0.0001\\ \hline \text{ones}& & \text{tenths} & \text{hundredths} & \text{thousandths} & \text{ten thousandths}\\ \hline \end{array}\]

Adding and Subtracting Decimals

Adding and subtracting decimals is pretty straightforward—you simply line up the decimal points. Add zeros to the right if you need to. Place the decimal point in the answer directly below where it appeared in the numbers you worked with.

Example 1: \(3.07 + 0.165\)
\(\begin {array}{r} & 3.070\\ +&0.165\\ \hline &3.235\\ \end{array}\)

Example 2: \(5-0.072\)
\(\begin {array}{r} &5.000\\ -&0.072\\ \hline &4.928\\ \end{array}\)

Multiplying Decimals

If you can’t use a calculator, multiply decimal numbers by following three steps:

  1. Multiply the two numbers ignoring decimal points.
  2. Count the total number of decimal places in both numbers.
  3. Count that many places from the right in the result and place the decimal point there.

Example: \(41.9 \times 6.2\)

\[419 \times 62 = 25,978\]

Count total decimal places (2). Place the decimal point two places from the right end.

\[259.78\]

Dividing Decimals

In case you have to divide decimals without a calculator, here are the steps:.

  1. Count the number of decimal places in the divisor (the number you’re dividing by).
  2. Move the decimal point that number of places to the right in the divisor, dividend, and quotient.
  3. Divide as usual.

Example: \(38 \div 6.25\)

Move the decimal points two places to the right in both numbers. Then place the decimal point directly above its landing spot in the quotient.

\[3800 \div 625\] \[\require{enclose} \begin{array}{r} 6.08\\[-3pt] 625\enclose{longdiv}{3800.00}\\[-3pt] \underline{3750}\phantom{.00}\\[-3pt] 5000\ \\[-3pt] \underline{5000}\ \\[-3pt] \end{array}\]

Rounding Numbers

When rounding, watch carefully to note the specified decimal place to which the answer is supposed to be rounded.

Decimal Places

Each decimal place represents a smaller and smaller fraction as you move to the right from the decimal point. Check out the table below for a refresher on the first five places.

\[\text {Decimal Place Names}\] \[\begin{array}{|c|c|c|c|c|} \hline & & & \text{Ten}&\text{Hundred}\\ \text{Tenths}&\text{Hundredths} &\text{Thousandths} & \text{Thousandths} & \text{Thousandths}\\ \hline 0.1 & 0.01 & 0.001 & 0.000\ 1 & 0.000\ 01\\ \hline \end{array}\]

Decimal numbers are named for their right-most digit.

\(0.05\) is five hundredths.
\(0.35\) is thirty-five hundredths.
\(0.009\) is nine thousandths.
\(0.425\) is four hundred twenty-five thousandths.

Rounding

The basic rule for rounding to a particular place value is to add \(1\) to that place only if the digit to the right is \(5\) or greater. If the digit to the right is not \(5\) or greater, do not add one. As an example, if you wanted to round to the nearest tenth, you would first look at the hundredths place. If there is a \(5\) or greater there, you would add \(1\) to the tenths digit. This is an example of rounding up.

Example 1

Round \(6.3274\) to the nearest hundredth.

The question we need to answer is this: Is the value of the given number closer to \(6.33\) or \(6.32\)? If the digit in the thousandths place is less than \(5\), the value is closer to \(6.32\), but if the thousandths digit is greater than \(5\), the value of the given number is closer to \(6.33\). The basic rule stated above comes from this reasoning. If the thousandths digit is \(5\) itself, we still round up.

So, looking at our example, we see to the right of the hundredths digit there is a \(7\). Since this is greater than \(5\), we add one to the \(2\) in the hundredths place, making it a \(3\), and drop the rest of the digits to the right. (Don’t write zeros in their place.)

Result: \(6.33\)
(Not \(6.3300\))

Example 2

Round \(74,249\) to the nearest hundred.

To the right of the hundreds place there is a \(4\). Since this is less than \(5\), we don’t add one to the \(2\) in the hundreds place. We leave it a \(2\) and drop the rest of the digits to the right, writing zeros in their place.

Result: \(74,200\)

Note: Don’t round \(74,249\) to \(74,250\) and then round that to \(74,300\) for your answer. Only round once, considering only the place you were asked to round to and the place immediately to its right.

Comparing Numbers

Sometimes a question will directly ask you to compare or order given numbers. Other times, it will just be a step along the way to solving a problem.

Converting Numbers

To compare numbers it’s best to have them in the same form. Very often, the best thing is to convert the numbers to their decimal form. For example, which is larger, \(\frac{8}{11} \text{ or }\frac{23}{31}\)? Their decimal equivalents tell the tale: \(0.727 \text{ vs. } 0.742\).

The other option is to change both fractions to a common denominator, which in some cases, may be easier than changing them to decimals, especially if you have no calculator. Many problems are designed to use simple addition and multiplication facts without a lot of calculation.

Whole Numbers and Fractions

If you’re working with a whole number and a fraction, maybe multiplying them, a good practice is to write the whole number as a fraction by writing it over \(1\). For example the number \(3\) would be written as \(\frac{3}{1}\).

To multiply \(3 \times \dfrac{2}{15}\):

\[\dfrac{3}{1} \times \dfrac{2}{15}\] \[\dfrac{3 \times 2}{1 \times 15}\] \[\dfrac{6}{15}\] \[\dfrac{2}{5}\]

It would be rare to have to convert a fraction to a whole number because few fractions would be equivalent to a whole number.

Fractions and Decimals

Fractions can be converted to decimals by dividing the denominator into the numerator and rounding the result to a reasonable number of decimal places. A calculator is our friend in this process, but if you’re not allowed to use one, you’ll just have to tough it out with long division.

Decimals can be turned into fractions by noticing place value and using that to rewrite the decimal as a fraction with a denominator of \(10\), \(100\), \(1000\), or some higher value. Then simplify the fraction.

Example: Convert \(0.24\) to a fraction.

\(0.24\) is \(24\) hundredths, and can be written as \(\dfrac{24}{100}\).

Simplify \(\dfrac{24}{100}\) and get \(\dfrac{6}{25}\).

Fractions and Percents

The word percent means per hundred, and it’s a way of standardizing ratios. For example, if one out of four people prefer only mustard on their hot dogs, that would be equivalent to saying that \(25\) out of \(100\) people prefer mustard. This would translate to \(25\) percent in both cases.

To convert a fraction to a percent, change the fraction to a decimal, as in the previous section, and move the decimal point two places to the right.

\[\dfrac{2}{5} = 0.40 = 40 \%\]

To convert a percent to a fraction, put the percent in a fraction as the numerator and make the denominator \(100\). Then simplify the fraction.

\[65 \% = \dfrac{65}{100} = \dfrac{13}{20}\]

Decimals and Percents

To convert a decimal to a percent, move the decimal point two places to the right and write the percent symbol to the number’s right.

Convert \(0.825\) to a percent.

\[0.825 = 82.5 \%\]

To convert a percent to a decimal, do just the opposite; move the decimal point two places to the left and drop the percent symbol.

Convert \(6.4 \%\) to a decimal.

\[6.4 \% = 0.064\]

Ordering Numbers

Ordering numbers is often best done by picturing them on a number line. Numbers to the right are always larger than numbers to the left. Remember, this means that \(-4\) is greater than \(-5\).

1-number-line-intro-visual.png

Retrieved from: https://openstax.org/books/elementary-algebra-2e/pages/1-3

If you need to order numbers of different types, say, fractions, decimals, and mixed numbers, and find it confusing, it’s usually best to convert everything to decimals. You might first look to see if you easily recognize a common multiple that you can use to convert to fractions with the same denominator.

Example 1

Order these numbers from smallest to largest: \(\frac{7}{8},\ 0.890,\ \frac{13}{15}\).

\[\dfrac{7}{8} = 0.875\] \[\dfrac{13}{15} = 0.867\]

The order of the decimals is \(0.867,\ 0.875, \text{ and } 0.890\).

That makes the order of the numbers in their original forms \(\frac{13}{15},\ \frac{7}{8},\ 0.890\).

Example 2

Place these numbers in descending order: \(3.6 \%, \ 0.0413, \ \dfrac{1}{27}, \ 0.2^2\).

\[3.6 \% = 0.036\] \[\dfrac{1}{27} = 0.037\] \[0.2^2 = 0.040\]

The descending order of the decimals is \(0.0413,\ 0.040, \ 0.037 \text{ and } 0.036\).

That makes the order of the numbers in their original forms \(0.0413,\ 0.2^2, \ \dfrac{1}{27}, \ 3.6 \%\).

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