Percentage above 100

Question

I am stuck on helping my nine-year old on a percentage question. She has quite a few questions of this form, however but I don't know the method of solving them. Here is one of the questions:

Question:

Yesterday $200$ flights left Heathrow airport. $21$ of these flights were to Edinburgh.
What percentage is this?

Possible Answers:

(A): $105.00\%$
(B): $10.50\%$
(C): $1.05\%$
(D): $21.00\%$
(E): $1050.00\%$

Can someone provide an explanation for this particular question so I can assist her on solving the others?

Update:

I apologize that I did not include what we have done thus far. Her and I are quite competent on working out $100\%$ questions. There was the following question which was fine for her:

"There are $25$ cars on the road and $18$ of them are red.
What percentage is this?

She simplified the ratio down to get $72\%$, which is the correct answer!

I think the problem is that I have no idea how to do anything about $100\%$. So when $200\%$ comes up I am lost on this. Any explanation would be appreciated. She currently does a lot of simplifying questions with percentages and I would assume this was similar but the books I have do not really explain anything above $100\%$ and I can't afford new books at the moment for her.

Answer 1

We want to find $x$ so that $\frac{21}{200}=\frac{x}{100}$

Multiplying by $100$ on both sides yields:

$x=\frac{100\cdot21}{200}=\frac{21}{2}=10.5$

Answer 2

It's hard to help you help your nine year old without a little more background about what you and she know so far. Here's a try.

"percent" comes from Latin and means "out of 100". In this problem you're given "21 out of 200". That's the same as "10 1/2 out of 100" so the correct answer is 10.5%.

It might help if you can see quickly that 20 flights out of 200 would be 10 of every 100, which would be 10 percent. Then you'd know the answer to this multiple choice question without having to do the work. It would have to be just over 10%.

The quick and dirty way to answer the question (which may not help in understanding percentages) is just to divide the given numbers and then move the decimal point two places to multiply by 100:

$$ \frac{21}{200} = 0.105 = 10.5\% .$$

This is clearly just a made up problem. Way more than 200 flights a day leave Heathrow. I doubt that 10% of them go to Scotland. How would you find out the real numbers?

Answer 3

this might be a great opportunity to talk about vocabulary in word problems.

Reword the question as "What part OF 200 IS 21 ?"

OF means "multiplied by"

IS means "is equal to"

So the question can again be translated as

"What number multiplied by 200 is equal to 21 ?"

setting up $$200\;x=21 \to x=\frac{21}{200} = 0.105$$

now go through the answers translating $105%$ as 105 PER CENT

PER means "divided by"

CENT means 100

so " 105% IS 105 divided by 100" . setting up ...

$$ 105 \% =\frac{105}{100}=1.05$$

Go through all the answers and see when the decimals match.

You might also mention that the construction OUT OF typically means "divided by" so that

$$10.5\% = 10.5 \text{ [ OUT OF ] } 100 = \frac{10.5}{100}$$

Answer 4

Percentage is a comfortable, standardized way to express the relation between two quantities in a multiplicative way.

This implies two dualities: absolute versus relative, and additive versus multiplicative.

First, think about absolute versus relative: one way how numbers appear in reality is as an absolute number, e.g., there are 14 cars parking, I have 45 dollars, I am 9 years old. Numbers can also appear in a relative sense, i.e., they express the size difference between two other numbers. There are two ways how to do this comparison: additively and multiplicatively, both explained in the following.

I am 9 years old, my cousin is 27 years old (both are absolute numbers). Now we compare them additively: the age of my cousin is 18 years larger than mine (=27-9). So the additive relative number 18 compares the 27 with the 9. Now multiplicatively: the age of my cousin is 3 times as large as mine (=27/9). So the multiplicative relative number 3 compares the 27 with the 9, too, but in a different, multiplicative sense. One can play with more examples to see how additive and multiplicative relations behave differently, e.g., if in 6 years the ages are 33 and 15. Further, for relative numbers one do not even need to know the two numbers that are compared! "Tina jumps 42 cm farer than Bob" does not reveal how far they actually jump, but only their additive relation.

Now, percentage is exactly the same as the multiplicative relative number, but simply multiplied by 100 ! So, the age of my cousin is 300% of mine, and the number 9 is 100% of 9, and the 9 is 150% of 6. And the 1 is 50% of 2.

Why should one multiply by 100? Well, it is easier to get a feeling on the size difference when the numbers are not within, say, $0...2$, but $0...200$. This holds particularly for smaller-relations, e.g., my dog's weight is $0.16$ times that of mine sounds/feels more complicated to think of, than my dog's weight is 16% of mine. Furthermore, one can think of 100 sticks, coins, etc. that represent one of the two entities to compare, and the other is then equal to, e.g., 170 sticks, or only 20 sticks. 100 is sufficiently large to think this way.

Unfortunately, there are (at least) two obstacles with multiplicative relative numbers like percentage (while additive relative numbers do not have these problems!). Sandy has 70% more puppets than Bob means that the number of puppets of Sandy is 170% of the number of puppets of Bob. And if an employee earns 4% more than before, than he now earns 104%. This subtle language difference on putting the percentage information on comparing the numbers directly (104%) versus expressing only their difference as a percentage (4%) of one of both is somehow tricky. A common mistake is to say "I have 130% more cookies than Bob", while meaning "I have 30% more cookies than Bob, namely 130% of the number of cookies of Bob". Another obstacle of multiplicative relations is that they are not "symmetric". In the additive comparison it is clear that if the cousin is 18 years older than you, than you are 18 years younger than the cousin. A similar thing does not hold for multiplicative relative numbers! Although Sandy has 70% more puppets than Bob, Bob has only 41.18% less puppets than Sandy! The tricky part is that in the first case 100% refers to Bob's number of puppets, while in the second part 100% refers to Sandys number of puppets. So if you now earn 4% more than before, then you earned before 3.85% less than now!

Better do not confuse your daughter with these obstacles until she learns the rule of proportion at school... even most adults run into problems in these subtle details :)

Answer 5

200 flights, 21 scheduled to Edinburgh. 21 out of 200 $\to (\frac{21}{200})*100$ for percentage = 10.5%

Or easier, 21 out of 200 fly to Edinburgh, so 10.5 out of 100 fly to Edinburgh, which is 10.5%

Answer 6

How about the rule of three for solving the proportions?

200 flights leave Heathrow airport $\to$ 21 flights are to Edinburgh

200 $\to$ 21

Divide both sides by 200:

1 $\to$ 0.105

Multiply both sides by 100:

100 $\to$ 10.5

So 10.5% is the correct answer.