Let's solve this ratio problem step-by-step to find the ratio of the number of marbles Joyee and Jacob had. We are given two initial ratios:

1. The ratio of marbles Joyee had to Minati had is 5 : 6.
2. The ratio of marbles Jacob had to Minati had is 5 : 9.

We want to find the ratio of the number of marbles that Joyee had to the number of marbles that Jacob had (Joyee : Jacob).

Understanding Ratios and Common Terms

A ratio compares two quantities. When dealing with multiple ratios involving common terms, we can use the common term to relate the others. In this case, Minati is the common person in both ratios.

Let $J$ be the number of marbles Joyee had, $M$ be the number of marbles Minati had, and $Ja$ be the number of marbles Jacob had.

From the given information, we can write the ratios as fractions:

• Joyee : Minati = 5 : 6 means $\frac{J}{M} = \frac{5}{6}$.
• Jacob : Minati = 5 : 9 means $\frac{Ja}{M} = \frac{5}{9}$.

Expressing Variables in Terms of the Common Term

To find the ratio of Joyee to Jacob ($J:Ja$), we need to express $J$ and $Ja$ in terms of the same variable, which is $M$ (Minati's marbles) in this case.

From the first ratio, $\frac{J}{M} = \frac{5}{6}$, we can solve for $J$:

$J = \frac{5}{6} M$

From the second ratio, $\frac{Ja}{M} = \frac{5}{9}$, we can solve for $Ja$:

$Ja = \frac{5}{9} M$

Calculating the Ratio of Joyee to Jacob

Now we can find the ratio of Joyee's marbles to Jacob's marbles, which is $\frac{J}{Ja}$. We substitute the expressions for $J$ and $Ja$ that we just found:

$\frac{J}{Ja} = \frac{\frac{5}{6} M}{\frac{5}{9} M}$

Since $M$ is a common factor in the numerator and the denominator, we can cancel it out (assuming Minati had more than 0 marbles, which is reasonable for a ratio problem):

$\frac{J}{Ja} = \frac{5/6}{5/9}$

To divide fractions, we multiply the numerator fraction by the reciprocal of the denominator fraction:

$\frac{J}{Ja} = \frac{5}{6} \times \frac{9}{5}$

Now, we can simplify by cancelling out the common factor of 5:

$\frac{J}{Ja} = \frac{\cancel{5}}{6} \times \frac{9}{\cancel{5}}$

$\frac{J}{Ja} = \frac{1}{6} \times \frac{9}{1}$

$\frac{J}{Ja} = \frac{9}{6}$

Finally, we simplify the fraction $\frac{9}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{J}{Ja} = \frac{9 \div 3}{6 \div 3} = \frac{3}{2}$

So, the ratio of the number of marbles that Joyee and Jacob had is 3 : 2.

Step-by-Step Solution Summary

1. Identify the given ratios: Joyee:Minati = 5:6 and Jacob:Minati = 5:9.
2. Identify the common term: Minati.
3. Express the quantities you want to compare (Joyee and Jacob) in terms of the common term Minati: $J = \frac{5}{6} M$ and $Ja = \frac{5}{9} M$.
4. Form the ratio of Joyee to Jacob: $\frac{J}{Ja}$.
5. Substitute the expressions in terms of M: $\frac{J}{Ja} = \frac{\frac{5}{6} M}{\frac{5}{9} M}$.
6. Simplify the complex fraction: $\frac{J}{Ja} = \frac{5}{6} \times \frac{9}{5}$.
7. Calculate and reduce the resulting fraction: $\frac{J}{Ja} = \frac{45}{30} = \frac{3}{2}$.

The ratio of Joyee's marbles to Jacob's marbles is 3 : 2.

Revision Table: Understanding Ratio Problems

Additional Information: Working with Three-Term Ratios

Another way to think about problems like this is to combine the ratios into a three-term ratio if possible. To do this, the common term must have the same "value" or proportion in both initial ratios.

We have Joyee : Minati = 5 : 6 and Jacob : Minati = 5 : 9.

Notice that Minati has different "parts" in the two ratios (6 and 9). To combine them, we need to find a common multiple for Minati's parts. The least common multiple (LCM) of 6 and 9 is 18.

Adjust the first ratio so Minati has 18 parts:

Joyee : Minati = 5 : 6

Multiply both parts by $\frac{18}{6} = 3$:

(5 $\times$ 3) : (6 $\times$ 3) = 15 : 18

So, Joyee : Minati is equivalent to 15 : 18.

Adjust the second ratio so Minati has 18 parts:

Jacob : Minati = 5 : 9

Multiply both parts by $\frac{18}{9} = 2$:

(5 $\times$ 2) : (9 $\times$ 2) = 10 : 18

So, Jacob : Minati is equivalent to 10 : 18.

Now we have Joyee : Minati = 15 : 18 and Jacob : Minati = 10 : 18. Since Minati has the same number of parts (18) in both adjusted ratios, we can relate Joyee and Jacob directly through Minati.

This means Joyee has 15 parts when Minati has 18 parts, and Jacob has 10 parts when Minati has 18 parts.

The ratio of Joyee to Jacob is therefore the ratio of their respective parts when Minati's part is common:

Joyee : Jacob = 15 : 10

Simplify this ratio by dividing both parts by their greatest common divisor, which is 5:

(15 $\div$ 5) : (10 $\div$ 5) = 3 : 2

This confirms the previous result. Both methods, using fractions or finding a common multiple for the connecting term, yield the same correct ratio for Joyee and Jacob.