Percentage and numeric relations

Question

I would like to know the simplest approach to the followings:

Que 1:

One student scored $14$ marks more than the other and his marks were $60$% of the sum of their marks. Find marks obtained by them.

My Approach:

Step 1: Assuming students scored $x$ and $x+14$.

Step 2: $x+14$ is $60$% of sum of their marks i.e. $2x+14$, so $x$ is $40$%.

Don't know further.

Ans: 28 and 42 I know the answer but don't know how to achieve.

 

Que 2:

Two persons are paid total $9.2$ Dollars per week. $1st$ is paid $260$% more than amount paid to $2nd$. Find their payment for week.

Ans: 2 and 7.2 I know the answer but don't know how to achieve.

Answer 1

A percentage is just a different way of writing a number. People do not like to say things like $0.4$, so instead they multiply the number by $100$ and attach a percentage sign to it. That is, by definition,

$$ 0.4 = 40\%.$$

People find it much more agreeable to say $40\%$ than to say $0.4$. They also prefer very much to say "$40\%$ of something" rather than "$40\%$ times something", although if you regard $40\%$ as a number (as I do) then the two phrases mean the same thing.

This is all fine until it comes time to actually do some arithmetic on the number. Then I personally find that usually the first thing I would like to do is to write $40\%$ as $0.4$, and say "times" rather than "of", and then never think about percentages again.

So for the first problem you found the following:

• The sum of the marks is $2x + 14$.
• $x$ is $0.4$ times the sum of the marks

If you introduce a symbol such as $s$ to represent "the sum of the marks", from the two statements above you can derive two equations which are easy to solve for $x$.

For the second problem, when people say something like "$x$ is $10\%$ more than $y$", they mean that $x$ is $y$ plus another $0.1$ times $y$, or $x = y + 0.1y = 1.1y$.

That is, "$10\%$ more than" is a way people like to say "$1.1$ times", "$20\%$ more than" is a way people like to say "$1.2$ times", "$50\%$ more than" is a way people like to say "$1.5$ times", and so forth.

So what does "$260\%$ more than" mean?

Now if $x$ is the wages of the lesser-paid worker, and the other worker is paid $260\%$ more than $x$, and the two wages add up to $9.2$, you can write a single easily-solved equation with one unknown $x$.

Answer 2

First question:

You are correct, $x$ is $40\%$ of the sum of their marks, and the sum of their marks is $2x+14$.

Now, you need to transform that into an equation.

In general, if $a$ is equal to $p\%$ of $b$ (for example, if "my height" is equal to $90\%$ of "your height"), you know that the ratio between $a$ and $b$, multiplied by $100$, is equal to $p$ (i.e., my height, divided by your height, times $100$, is $90$).

Now, answer these questions (in that order):

• how would you write the relation "ratio between $a$ and $b$, multiplied by $100$, is equal to $p$" as an equation?
• In your case, what is $a$ and what is $b$, and what is $p$?
• So in your case, what equation do you get?
• How do you solve this equation?

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Second question:

Say the first person is paid $x$, and the second one is paid $y$. Then, transform the following two statements into equations:

• They are paid $9.2$ dollars in total.
• The pay of the first person is $260\%$ more than the pay of the second person.

What equations do you get?