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Ratios & proportions: Combining Ratios with Different Multipliers - Equate the Common Member

A bottle contains a certain solution. In the bottled solution, the ratio of water to soap is $$3{:}2$$, and the ratio of soap to salt is 3 times this ratio. The solution is poured into an open container, and after some time, the ratio of water to soap in the open container is halved by water evaporation. At that time, what is the ratio of water to salt in the solution?



       water    :    soap    :    salt

1st ratio            ( 3    :    2 )

Multiply the water:soap ratio by 3 to find the $$soap{:}salt$$ ratio. A $$3{:}2$$ ratio multiplied by 3 is $$9{:}2$$ (remember, multiplying or dividing a ratio demands multiplying or dividing the left term only).

2nd  ratio                      ( 9      :        2 )

Next, calculate the new $$water{:}soap$$ ratio resulting from the evaporation. The new ratio is half of the old one; $$3{:}2$$ divided by 2 is $$1.5{:}2$$. Expand the resulting ratio by 2 to get rid of the decimal, and get $$3{:}4$$.

New 1st ratio:  ( 3    :   4 )

Since the common member of both ratios, soap, is represented by different numbers, combine the two ratios by expanding both of them to equate the common member. Expand the first ratio by 9, and the second ratio by 4, to get a common ratio number of 36.

Diagram of the equate ratios:

      water       :       soap        :        salt

1st ratio     ($$\color{red}{3}\times9=\color{purple}{27}$$    :   $$\color{red}{4}\times9=36$$)

2nd  ratio                             ($$\color{red}{9}\times4=36$$   :   $$\color{red}{2}\times4=\color{purple}{8}$$)

The ratio of $$water{:}salt$$ is therefore $$\color{purple}{27{:}8}$$.

To sum this up:

$$2{:}3$$ multiplied by 5 results in the new ratio $$10{:}3$$.

$$2{:}3$$ expanded by 5 results in $$10{:}15$$, which is the same ratio—the change is only cosmetic.


Okay. Let's move on.

A ratio is like a fraction: a ratio of $$4{:}3$$ is mathematically equivalent to the fraction $$\frac{4}{3}$$. Thus, multiplying or dividing a ratio is like multiplying or dividing a fraction: you need to multiply or divide only the left number (which is equivalent to the numerator), and not both numbers.

For example:

A ratio of $$4{:}3$$ multiplied by 2 becomes $$(4\times2){:}3 = 8{:}3$$.

A ratio of $$4{:}3$$ divided by 2 becomes $$(\frac{4}{2}){:}3 = 2{:}3$$

Multiplying or dividing both numbers in a ratio is equivalent to expanding or reducing a fraction, not multiplying/dividing it. Exactly as with fractions, reducing or expanding a ratio doesn't change its value. Multiplying or dividing a ratio, on the other hand, result in an entirely new ratio.

For example, the ratio $$2{:}6$$ is equal to $$1{:}3$$ , just as $$\frac{2}{6}$$ is equal to $$\frac{1}{3}$$.

Why do we only multiply or divide only the left term when multiplying or dividing a ratio?
Got it!