Max and Tanya are painting two acjacent walls of equal area. Max has painied & of his wall and

Tanya has painted 2
of her wall.
a What fraction of the two wal have Max and Tanya painted in total? b What fraction of the two walls remains to be painted?

1 answer

Let \( A \) be the area of each wall. Since Max and Tanya are painting two walls of equal area, the total area of both walls combined is:

\[ 2A \]

a. Max has painted \(\frac{1}{3}\) of his wall, so the area he has painted is:

\[ \text{Area painted by Max} = \frac{1}{3}A \]

Tanya has painted \(\frac{2}{5}\) of her wall, so the area she has painted is:

\[ \text{Area painted by Tanya} = \frac{2}{5}A \]

Now, we find the total area painted by both Max and Tanya:

\[ \text{Total area painted} = \frac{1}{3}A + \frac{2}{5}A \]

To add these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15. So, we convert each fraction:

\[ \frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{2}{5} = \frac{6}{15} \]

Adding these together:

\[ \frac{5}{15}A + \frac{6}{15}A = \frac{11}{15}A \]

Thus, the total area painted by Max and Tanya in terms of the area of both walls is:

\[ \text{Fraction of total area of both walls painted} = \frac{11}{15}A \div 2A = \frac{11}{30} \]

So, the fraction of the two walls that Max and Tanya have painted in total is:

\[ \boxed{\frac{11}{30}} \]

b. Now, we need to find the fraction of the two walls that remains to be painted. The total area of the two walls is \(2A\), and the area painted is \(\frac{11}{15}A\) as calculated above.

The area that remains to be painted is:

\[ \text{Area remaining} = 2A - \frac{11}{15}A \]

To subtract these, we convert \(2A\) into a fraction with the same denominator:

\[ 2A = \frac{30}{15}A \]

Now we can subtract:

\[ \frac{30}{15}A - \frac{11}{15}A = \frac{19}{15}A \]

Now we find the fraction of the total area of both walls that remains to be painted:

\[ \text{Fraction remaining} = \frac{19}{15}A \div 2A = \frac{19}{30} \]

So, the fraction of the two walls that remains to be painted is:

\[ \boxed{\frac{19}{30}} \]