Math Trailhead

Solution

The mode ( the number that occurs most frequently) is 280 and will not change

The median (the central number) is 280 and will change to 279.5 (the average of 280 and 279),

Since the mean (the average of the numbers) is about 280 and we add 101 which is about 180 less

than the mean, and there are now 10 values, the mean will decrease by approximately \(180/10=18\text{.}\)

Solution

Notice that the price changes by $ 1.35 when one goes down one row (for each of the first three rows].

It changes by .05 [which is \(3\times 1.35\)] when two rows are skipped. Thus it will

be .35 more than the last row in the 25-29 row. So the price is $ \(5.90+1.35=6.25\text{.}\)

Row 1 To convert a percent into a fraction, divide it by 100 and reduce.

\(4\% = \frac{4}{100} = {{\frac{1}{25}}}\)

To convert a percent into a decimal, move the decimal two places to the left.

\(4\% = {0.04}\)

Row 2 To convert a fraction into a percent, we can set up the proportion:

\({{\frac{19}{20}}} = \frac{p}{100}\)

\(p = {{\frac{19}{20}}} \cdot 100 = {95}\%\)

Now we can convert the percent to a decimal by moving the decimal two places to the left.

\({95}\% = {0.95}\)

Row 3 To convert a decimal to a percent, move the decimal two places to the right.

\(2.89 = {289}\%\)

Put the percent over 100 (and reduce) to convert it to a fraction.

\(\frac{{289}}{100} = {{\frac{289}{100}}}\)

Summary

\(4\% = {{\frac{1}{25}}} = {0.04}\)

\({95}\% = {{\frac{19}{20}}} = {0.95}\)

\({289}\% = {{\frac{289}{100}}} = 2.89\)

Solution

Notice that when there are 4 grams of peanut butter there is 1 gram of protein.

Thus there is \(\frac{1}{4}\) gram of protein in 1 gram of peanut butter.

Find the mean

To find the mean of a group of numbers, we first add up all numbers and find their sum:

\(\displaystyle{ \text{sum} = 12+19+3+11+17+10+1+16+19 = 108 }\)

Next, we divide the sum by how many numbers there are:

\(\displaystyle{ \text{mean} = \frac{108}{9} = 12 }\)

The mean of this group of numbers is \(12\text{.}\)

Find the median

To find the median of a list of number, we first need to order these numbers from smallest to largest:

\(\displaystyle{ 1,\;\;3,\;\;10,\;\;11,\;\;12,\;\;16,\;\;17,\;\;19,\;\;19 }\)

The number in the middle is the median.

The median of this group of numbers is \(12\text{.}\)

The mean is the average of the numbers. To find the mean add the values and then divide by the number of values (5).

Mean = \((6 + 4 + 6 + 10 + 12) \div 5\)

Mean = \(38 \div 5\)

Mean = \(7.6\)

The median is the middle number. First put the values in order, from smallest to largest:

\(4, 6, 6, 10, 12\)

The middle number is the 3rd value, 6.

The mode is the vale that occurs most frequently. Since there are two 6’s, the mode is: 6.

This problem can be boiled down to this question: What is \(22\%\) of \(1950\text{?}\) We will show multiple methods to solve this problem.

Method 1

We will use proportion to solve this problem. Assume \(22\%\) of \(1950\) is \(x\text{,}\) so “\(22\) out of \(100\)” corresponds to “\(x\) out of \(1950\)”.

We will write and solve the proportion:

\(\displaystyle{\begin{aligned}[t] \frac{22}{100} \amp = \frac{x}{1950} \\ 100x \amp = 22 \cdot 1950 \\ 100x \amp = 42900 \\ \frac{100x}{100} \amp = \frac{42900}{100} \\ x \amp = 429 \end{aligned} }\)

There are \(429\) signatures by the Beatles in this collection.

Method 2

We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

\(2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4\)

The question is: What is \(22\%\) of \(1950\text{?}\) Assume \(x\) is \(22\%\) of \(1950\text{.}\) We have:

\(\displaystyle{ \begin{aligned} x \amp = 0.22 \cdot 1950 \\ \amp = 429 \end{aligned} }\)

There are \(429\) signatures by the Beatles in this collection.

Method 3

In the sentence “What is \(22\%\) of \(1950\text{,}\)”

By the formula \(\text{percentage} = \text{rate} \cdot \text{base}\text{,}\) we do a multiplication to solve the problem:

\(\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 22\% \cdot 1950 = 0.22 \cdot 1950 = 429 }\)

There are \(429\) signatures by the Beatles in this collection.

We can use subtraction to find the percent of Sting signatures:

\(\displaystyle{\begin{aligned}[t] \amp \phantom{{}=}1-34\%-20\%-14\%-10\% \\ \amp =1-0.34-0.2-0.14-0.1 \\ \amp =0.22 \\ \amp = 22\% \end{aligned} }\)

So there are \(22\%\) Sting signatures. Now this problem can be boiled down to this question: What is \(22\%\) of \(450\text{?}\) We will show multiple methods to solve this problem.

Method 1

We will use proportion to solve this problem. Assume \(22\%\) of \(450\) is \(x\text{,}\) so “\(22\) out of \(100\)” corresponds to “\(x\) out of \(450\)”.

We will write and solve the proportion:

\(\displaystyle{\begin{aligned}[t] \frac{22}{100} \amp = \frac{x}{450} \\ 100x \amp = 22 \cdot 450 \\ 100x \amp = 9900 \\ \frac{100x}{100} \amp = \frac{9900}{100} \\ x \amp = 99 \end{aligned} }\)

There are \(99\) signatures by Sting in this collection.

Method 2

We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

\(2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4\)

The question is: What is \(22\%\) of \(450\text{?}\) Assume \(x\) is \(22\%\) of \(450\text{.}\) We have:

\(\displaystyle{ \begin{aligned} x \amp = 0.22 \cdot 450 \\ \amp = 99 \end{aligned} }\)

There are \(99\) signatures by Sting in this collection.

Method 3

In the sentence “What is \(22\%\) of \(450\text{,}\)”

By the formula \(\text{percentage} = \text{rate} \cdot \text{base}\text{,}\) we do a multiplication to solve the problem:

\(\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 22\% \cdot 450 = 0.22 \cdot 450 = 99 }\)

There are \(99\) signatures by Sting in this collection.