# Ratio problem

• Mar 22nd 2010, 12:02 PM
Natasha1
Ratio problem
My child needs help on this problem please...

In a survey, the ratio of the number of people who preferred MILK CHOCOLATE to those who preferred PLAIN CHOCOLATE was 5:3

46 more people preferred MILK CHOCOLATE, to PLAIN CHOCOLATE.

How many people were in the survey?

What she's done so far is 5+3 = parts in the ratio and said that there are 64 people who took part in the survey as 8x5 = 40 and 8x3 = 24

Many thanks!
• Mar 22nd 2010, 12:12 PM
e^(i*pi)
Quote:

Originally Posted by Natasha1
My child needs help on this problem please...

In a survey, the ratio of the number of people who preferred MILK CHOCOLATE to those who preferred PLAIN CHOCOLATE was 5:3

46 more people preferred MILK CHOCOLATE, to PLAIN CHOCOLATE.

How many people were in the survey?

What she's done so far is 5+3 = parts in the ratio and said that there are 64 people who took part in the survey as 8x5 = 40 and 8x3 = 24

Many thanks!

Let the number that like milk chocolate be $\displaystyle x$
Let the number that like plain chocolate be $\displaystyle y$

A ratio means the value of $\displaystyle \frac{x}{y}$. The ratio of five to three in favour of milk means:

$\displaystyle \frac{x}{y} = \frac{5}{3}$ and hence $\displaystyle 3x = 5y$ (eq1)

If 46 more people preferred milk to plain then $\displaystyle x-y = 46$

There are now two equations and two unknowns, has your daughter done simultaneous equations?
• Mar 22nd 2010, 12:14 PM
Natasha1
She hasn't. She's only 11. And needs a clear, simple understanding.
• Mar 22nd 2010, 12:31 PM
e^(i*pi)
Quote:

Originally Posted by Natasha1
She hasn't. She's only 11. And needs a clear, simple understanding.

Take 2 :)

Using the ratio we can see that for every $\displaystyle 5+3 = 8$ people then 5 prefer milk chocolate and 3 prefer plain chocolate.

Let there be $\displaystyle x$ people.

$\displaystyle \frac{5x}{8}$ prefer milk chocolate

$\displaystyle \frac{3x}{8}$ prefer plain chocolate

We are told that the difference between these numbers is equal to $\displaystyle 46$

$\displaystyle \frac{5x}{8} - \frac{3x}{8} = 46$

We can simplify the LHS

$\displaystyle \frac{2x}{8} = \frac{x}{4} = 46$

$\displaystyle x = 46 \times 4 = 184$

Therefore $\displaystyle 184 \times \frac{5}{8} = 115$ students preferred milk chocolate whereas $\displaystyle 115 \times \frac{3}{8} = 69$ preferred plain.

This can be verified: if $\displaystyle 115-69 = 46$ is true then the answers are fine. Since this is the case the answer is correct
• Mar 22nd 2010, 12:36 PM
Natasha1
Can this whole exercise be done without simultaneous equations?
• Mar 22nd 2010, 01:03 PM
e^(i*pi)
See my second post - that only uses one variable
• Mar 23rd 2010, 12:40 AM
sandrodacomo
By thinking, no algorithms
Hi Natasha.

The teacher wants to make children think on (the very difficult concept of ) ratios and proportional thinking.

So, let the child try to think, without urging for a result which might not even come. The exercise is about thinking not about getting a result (succeed). And the child must make drawings!

Here is my thinking. I begin always with easy numbers...

The original ratio is 5 to 3. In our example 5 people liked "this" and 3 people liked "that". Or, I can state that there were 8 people and 5 of them liked milk chocolate and 3 liked plain chocolate. (make a drawing with "dots")

5 and 3 people. So, I can see the difference: 2 people more liked milk chocolate.

Now the problem tells me that in the survey there are more than 8 people , right? Right! So how many can I add to keep the proportion?

Well, try it!

I double the people and the get 10 and 6, I double again and they get 20 and 12, and so on...(these are equivalent fractions by the way, but it's not important to know the name). The idea is that I can change the numbers of the people in the groups without changing their relative proportion.

But wait again, there is an easier way! I know that with 8 people and the proportion 5 to 3 there are 2 people more in one group!

Ok! Now the problem says that there are 46 people more in one group! How many more times is the difference of people between the groups? 23! (Why? Well, 23 times the difference of 2 makes 46!)

So now we have the key for the solution: 23!

We can change the original people by multiplying them by 23: and we get 115 (5x23) and 69 (3x23). You can also see it here:

(START) 5 and 3, difference is 2 people
(TIMES 2) 10 and 6, difference is 4 people
(TIMES 3) 15 and 9, difference is 6 people
(TIMES 4) 20 and 12, difference is 8 people
......
(TIMES 23) 115 and 69, difference is 46

So there is a total of 115+69=184 people in the survey.

The idea of the teacher is probably to make an introduction into equivalent fractions.

I know fractions are not easy (including for myself of course!) because proportional thinking is not intuitive at all. I am making a website on it (www.swiss-algebra-help.com) and the more I write the more surprises I find...

Alessandro
Switzerland
• May 8th 2010, 03:12 AM