To get the apple to carrot ratio, we need to equal out the bananas. The least common denominator of \(\displaystyle 2\) and \(\displaystyle 3\) is \(\displaystyle 6\). So if \(\displaystyle 3\) apples equal \(\displaystyle 2\) bananas, then \(\displaystyle 6\) bananas equal \(\displaystyle 9\) apples. Also, if \(\displaystyle 3\) bananas equal \(\displaystyle 4\) carrots, then \(\displaystyle 6\) bananas equal \(\displaystyle 8\) carrots. Since now the total bananas are equal, we can find the ratio of apples to carrots. We have \(\displaystyle 9:8\) as the final answer.

We can rewrite the ratio \(\displaystyle 0.57:0.95\) as a fraction. The first number in the ratio is in the numerator while the second number in the ratio is in the denominator.

\(\displaystyle \frac{0.57}{0.95}=\frac{57}{95}=\frac{3}{5}\) Remember to get rid of decimals, we can move the decimal point two places to the right. Afterwards, the two numbers are divisible by \(\displaystyle 19\).

To find an integer ratio, let's find the fractions with a common denominator. This will be \(\displaystyle 6\). Then, we multiply the left by \(\displaystyle 3\) and the right by \(\displaystyle 2\) to get fractions of \(\displaystyle \frac{3}{6}\) and \(\displaystyle \frac{2}{6}\). With the same denominators, we just have numerators to compare. Ratio is then \(\displaystyle 3:2\).

If there are \(\displaystyle 7\) girls and \(\displaystyle 6\) boys, that means we have \(\displaystyle 13\) students in the class. To continue to have this ratio, we need an answer than is a multiple of \(\displaystyle 13\). 

\(\displaystyle 39\) is a multiple of \(\displaystyle 13\) which is the right answer.

In the problem, the drink is \(\displaystyle 30\%\) espresso since the overall weight of the drink is \(\displaystyle 10\) ounces. If we are reducing the concentration of espresso to \(\displaystyle 25\%\), then we can create an equation to figure out the addition of water.

\(\displaystyle \frac{3}{10+x}=\frac{1}{4}\) \(\displaystyle x\) represents the addition of water.

Cross-multiply.

\(\displaystyle 12=10+x\) 

Subtract \(\displaystyle 10\) on both sides. 

\(\displaystyle x=2\)

If Jill, Jack and John get \(\displaystyle 6:9:5\), that means there are \(\displaystyle 20\) parts.

Because they found \(\displaystyle 1000\), each part gets \(\displaystyle \frac{1000}{20}\) or \(\displaystyle \$50\).

Jack gets \(\displaystyle 50\cdot9\) or \(\displaystyle \$450\).

John gets \(\displaystyle 50\cdot5\) or \(\displaystyle \$250\).

Since the question is asking how much more did Jack get than John, we subtract \(\displaystyle 450\) and \(\displaystyle 250\) to get \(\displaystyle 200\).

You don't need to solve for the actual number of broken or unbroken dolls. Instead, put the percentages in the ratio because no matter what, the percentages are fixed regardless of amount of dolls broken or unbroken.

So the question is asking for broken to unbroken. The percentage of broken dolls is \(\displaystyle 4\%\).

So we have a ratio of \(\displaystyle 4:96\) or \(\displaystyle 1:24\). 

We need to understand that there is a total of \(\displaystyle 24\) ounces of solution in a pot. Out of that solution, \(\displaystyle 6\) ounces is sugar. If we add \(\displaystyle 4\) ounces of sugar, we are also changing the volume of the pot. There is a total of \(\displaystyle 10\) ounces of sugar and a total volume of \(\displaystyle 28\) ounces.

To find percentage, we do \(\displaystyle \frac{10}{28}\cdot 100\%\) which is \(\displaystyle 35.7\%\).

Since the problem is focusing on Carol, we will only worry about Carol's contribution to the bill.

The original split on the bill was \(\displaystyle 3:2:1\) with Carol paying the smaller portion. With \(\displaystyle 6\) parts, each one was \(\displaystyle \$10\). This was originally the amount Carol was responsible for.

With the the ratio of \(\displaystyle 2:3:5\), this meant there are \(\displaystyle 10\) parts with each being \(\displaystyle \$6\). Carol pays instead \(\displaystyle \$30\).

The difference is then \(\displaystyle 30-10\) or \(\displaystyle \$20\).