Why they multiply in this case?

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Edin

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Hello, I have solutions to these 2 math tasks but I am very confused about why they multiply in these 2 cases. May you please help me understand what I miss?

1st math task:
A baker uses 0.6 ton of whole wheat flour each month. 1/9 of the whole wheat flour is used for muffins. If the baker adds 0.1 ton of all-purpose flour to the muffin miss, how much does the muffin mix weight?
The solution is (6/10x1/9) +1/10 = Answer 1/6

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But why, if the baker use 1/9 of 0.6 ton I think that I have to divide 0.6 ton of 1/9, because the 0.6 ton of the whole wheat flour is going less, right? I don't get it....

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2nd math task:
A scientist has a 1.5 liter bottle of water. He pours 9/10 of the bottle into a beaker and boils it so 0.25 liter of the liquid evaporates. How much water is left in the beaker?

The solution is again multiplication: (1.5 x 9/10) - 0.25= Ans: 1 1/10
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The water from the bottle is less when he took 9/10 from the water, so we have to do in my opinion 1.5 - 9/10, why the multiplication is applied again for finding a solution?

Please kindly help me to understand the logic in these 2 tasks. I am very confused about why in these both cases there have to be applied a multiplication when it is obviously the quantities in both case is going less, so I think I have divided or put a minus. I was trying to understand, but I am still very confused.

Thank you for your time and help.
 
A scientist has a 1.5 liter bottle of water. He pours 9/10 of the bottle into a beaker and boils it so 0.25 liter of the liquid evaporates. How much water is left in the beaker?

9/10 is less than 1. If you divide a positive number by a positive number less than 1, then the number grows. For example, 10/(1/2) = 20.

If you compute 1.5/(9/10) you get 5/3.

Lets use friendlier numbers. A scientist has a 4 liter bottle of water. He pours 1/2 of the bottle into a beaker. How much water idid he pour into the beaker?

Please post back with your work.
 
Edin said:
A baker uses 0.6 ton of whole wheat flour each month. 1/9 of the whole wheat flour is used for muffins. If the baker adds 0.1 ton of all-purpose flour to the muffin miss, how much does the muffin mix weight?
The solution is (6/10x1/9) +1/10 = Answer 1/6
Click to expand...

Well, 1/9 OF 0.6 ton means 1/9 TIMES 6/10 ton. That is the same as 6/10 ton DIVIDED BY 9, not divided by 1/9. You have made a very common beginner's error here.

Think about it. 1/9 of 6/10 ton has to be less than 6/10 ton, as you say? Which calculation, [MATH](1/9)\times(6/10)[/MATH] or [MATH](6/10)\div(1/9)[/MATH], results in a smaller number? Carry them both out, and see.
Edin said:
A scientist has a 1.5 liter bottle of water. He pours 9/10 of the bottle into a beaker and boils it so 0.25 liter of the liquid evaporates. How much water is left in the beaker?
Click to expand...
First, the 9/10 is poured into the beaker, not out of it, so there is no subtraction. We are not asked how much is left in the bottle, but in the beaker!

Second, the amount poured is not 9/10 of a liter, but 9/10 of the bottle, that is, 9/10 of the 1.5 liters. That is multiplication.
 
A critical word here is "of". "1/9 of the whole wheat flour is used for muffins." "He pours 9/10 of the bottle into a beaker and boils it so 0.25 liter of the liquid evaporates." A fraction of something is always that fraction multiplied by the amount.

You say "But why, if the baker use 1/9 of 0.6 ton I think that I have to divide 0.6 ton of 1/9, because the 0.6 ton of the whole wheat flour is going less, right? I don't get it...."

Yes, and that is why we need to multiply by a fraction less than one! Do you understand that multiplying by 1/9 is the same as dividing by 9? 1/9 times 0.6 is (1/9)(0.6)= 0.6/9= 0.0666... which is much less than 0.6. If we were to divide by 1/9, that is the same as multiplying by 9 which is much 0.6(9)= 5.4, much larger.
 
Thank you very much, Jomo, Dr.Peterson and HallsofIvy. Each of you helps me a lot. I understand more completely when I read all 3 posts. I need it these 3 different ways of explanation. I will redo again and again these math cases in order to understand this concept.
 
I redid again these math tasks. I really appreciate Jomo, HallsofIvy and HallsofIvy your 3 helpful and very valuable ways of explanation that says the same concept but in different ways and that helps me to build the missing part of the puzzle in my head. I didn't realize what I was actually missing. Now I think that I get it and I will try to apply for similar math tasks in the future.

Jomo wrote:
"Lets use friendlier numbers. A scientist has a 4 liter bottle of water. He pours 1/2 of the bottle into a beaker. How much water idid he pour into the beaker?"

Wow, I understood even more now after this example. I am very grateful, Jomo! Thank you very much!

I have calculated right away in my head that 4/2 = 2, so 2 litres he poured into a beaker. I see that because I know this whole number very well I didn`t divide by 1/2 but by 2 because I know that 1/2 of 4 is 2.

But if I do a proper calculation in my wrong previous logic I have to take 4 and divide by 1/2, so in this case, I will have:
4 ÷ 1/2= 4 x 2/1 = 6 litres and it will become a bigger number than the bottle.

Thank you very very very much for taking your time and help me!
 
Ops, sorry, typing error. 4 ÷ 1/2= 4 x 2/1 = 8, not 6. Sorry about that.
 
Whenever you are confused about some concept never use unfriendly numbers, just change them temporarily to friendly numbers figure out what to do and then go back and do the same procedure to the unfriendly numbers>
 
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