Ratio and Percentages

Question

I have two questions related to ratios and percentages:

1. What's the best way to work out the value of a unit if the ratio changes? Only example I could think of:

"If you have two bags of marbles with the ratio: "Bag A $3:5$ Bag B". Now if you remove 5 marbles from bag B and put them in bag $A$, the new ration is: "Bag A $7:9$ Bag B"."

2. If $5$ is $17\%$, how do I then work out the value of $83\%$ left over and then I can establish the total value.

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P.S.: Sorry if these seems simple but struggling with it.

Answer 1

In the Singapore Primary School math system, a lot of emphasis is placed on model drawing without explicitly using algebra. It's possible to solve your first problem this way.

Initial ratio $= 3:5$ Total number of units = $3+5 = 8$

An internal transfer of $5$ objects now occurs.

Final ratio $=7:9$. Total number of units = $7+9 = 16$

Since there has been no external transfer, you can compare the two by getting the total number of units to be the same. The initial ratio is multiplied by two to give $6:10$, which has a total of $16$ units, equal to the final ratio.

Clearly, $1$ unit has been transferred between the two sides of the ratio to go from the initial to the final, and that's equivalent to $5$ marbles.

So the total of $16$ units is equivalent to $(16)(5)=80$ marbles. [answer]

For your second problem, if $5$ is equivalent to $17\%$, then $1\%$ is $\frac{5}{17}$. You can immediately get the total ($100\%$) from this by multiplying by $100$, so the total is $\frac{(5)(100)}{17} \approx 29.41$. But if you really wanted $83 \%$, it is $\frac{(5)(83)}{17} \approx 24.41$

Answer 2

1) Let $n_a$ and $n_B$ be the number of marbles in bag $A$ and $B$, respectively.

Now, you know that the ratio of marbles is initially some number $$R_i=\frac{n_A}{n_B}.$$ Now say you remove $m$ (in your case $m=5$) from $B$ and put them in $A$, and you know the final ratio $R_f$ for which

$$R_f=\frac{n_A+m}{n_B-m}.$$

Thus you have two equations with two unknowns which can be solved to give

$$n_A=mR_iK$$ and $$n_B=mK,$$

where $K=\frac{R_f+1}{R_f-R_i}$.

2) If it is simply the total number of marbles $n_A+n_B$ you're after, use the above equations to get $$n_A+n_B=mK(R_i+1).$$ Alternatively (and easier), you can simply write $$\text{Percentage}=\frac{m}{n_A+n_B}\implies 17\%=\frac{5}{n_A+n_B}\implies n_A+n_B=\frac{5}{0.17}\approx 29.$$