In a town, 85% of the people speak Tamil, 40% speak English, and 20% speak Hindi. 32% speak Tamil and English, 13% speak Tamil and Hindi, and 10% speak English and Hindi. We need to find the percentage of people who can speak all three languages using a Venn diagram.

1. Problem Description

In a town, 85% of the people speak Tamil, 40% speak English, and 20% speak Hindi. 32% speak Tamil and English, 13% speak Tamil and Hindi, and 10% speak English and Hindi. We need to find the percentage of people who can speak all three languages using a Venn diagram.

2. Solution Steps

Let T be the percentage of people who speak Tamil.

Let E be the percentage of people who speak English.

Let H be the percentage of people who speak Hindi.

Let $T \cap E$ be the percentage of people who speak Tamil and English.

Let $T \cap H$ be the percentage of people who speak Tamil and Hindi.

Let $E \cap H$ be the percentage of people who speak English and Hindi.

Let $T \cap E \cap H$ be the percentage of people who speak all three languages.

We are given:

$T = 85\%$

$E = 40\%$

$H = 20\%$

$T \cap E = 32\%$

$T \cap H = 13\%$

$E \cap H = 10\%$

Let $x$ be the percentage of people who speak all three languages, $T \cap E \cap H = x$.

The percentage of people who speak Tamil and English, but not Hindi, is:

$T \cap E - x = 32\% - x$

The percentage of people who speak Tamil and Hindi, but not English, is:

$T \cap H - x = 13\% - x$

The percentage of people who speak English and Hindi, but not Tamil, is:

$E \cap H - x = 10\% - x$

The percentage of people who speak only Tamil is:

$T - (T \cap E) - (T \cap H) + x = 85\% - 32\% - 13\% + x = 40\% + x$

The percentage of people who speak only English is:

$E - (T \cap E) - (E \cap H) + x = 40\% - 32\% - 10\% + x = -2\% + x$

The percentage of people who speak only Hindi is:

$H - (T \cap H) - (E \cap H) + x = 20\% - 13\% - 10\% + x = -3\% + x$

The sum of the percentages of people speaking each language is 100%.

$(40\% + x) + (-2\% + x) + (-3\% + x) + (32\% - x) + (13\% - x) + (10\% - x) + x \le 100\%$

$40\% + x - 2\% + x - 3\% + x + 32\% - x + 13\% - x + 10\% - x + x \le 100\%$

$90\% + x \le 100\%$

$x \le 10\%$

Consider the Inclusion-Exclusion Principle:

$T \cup E \cup H = T + E + H - (T \cap E) - (T \cap H) - (E \cap H) + (T \cap E \cap H)$

$T \cup E \cup H = 85\% + 40\% + 20\% - 32\% - 13\% - 10\% + x$

$T \cup E \cup H = 90\% + x$

Since the total percentage cannot exceed 100%, we have:

$90\% + x \le 100\%$

$x \le 10\%$

Let's consider the English speakers. The percentage speaking only English must be non-negative.

$-2\% + x \ge 0$

$x \ge 2\%$

Also, the percentage speaking only Hindi must be non-negative.

$-3\% + x \ge 0$

$x \ge 3\%$

So we have $3\% \le x \le 10\%$.

If $x=3\%$,

$T \cup E \cup H = 90\% + 3\% = 93\%$

Only Tamil $= 40\% + 3\% = 43\%$

Only English $= -2\% + 3\% = 1\%$

Only Hindi $= -3\% + 3\% = 0\%$

Tamil and English only $= 32\% - 3\% = 29\%$

Tamil and Hindi only $= 13\% - 3\% = 10\%$

English and Hindi only $= 10\% - 3\% = 7\%$

All three $= 3\%$

Sum $= 43\% + 1\% + 0\% + 29\% + 10\% + 7\% + 3\% = 93\%$

We need to find a specific value for $x$. Assume all values are given precisely so as to yield a single value for $x$. This means that if 100% of people can speak at least one language, then we have $90\%+x=100\%$, $x=10\%$.

If $x=10\%$,

$T \cup E \cup H = 90\% + 10\% = 100\%$

Only Tamil $= 40\% + 10\% = 50\%$

Only English $= -2\% + 10\% = 8\%$

Only Hindi $= -3\% + 10\% = 7\%$

Tamil and English only $= 32\% - 10\% = 22\%$

Tamil and Hindi only $= 13\% - 10\% = 3\%$

English and Hindi only $= 10\% - 10\% = 0\%$

All three $= 10\%$

Sum $= 50\% + 8\% + 7\% + 22\% + 3\% + 0\% + 10\% = 100\%$

3. Final Answer

10%