The following table of grouped data represents the weight (in kg) of $100$ gas cylinders. Calculate the mode weight of a cylinder.
Weight $(%)$ $3 - 5$ $5 - 7$ $7 - 9$ $9 - 11$ $11 - 13$ $13 - 15$
Number of gas cylinders $16$ $13$ $15$ $25$ $14$ $17$

12.96

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15.96

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9.96

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10.96

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Solution

The correct option is D $9.96$
The lower limit ( $l$ ) of the modal class $= 9$ ,
the class size ( $h$ ) $= 2,$
the frequency $(f_{1})$ of modal class $= 25,$
the frequency $(f_{0})$ of the class preceding the modal class $= 15$ ,
the frequency $(f_{2})$ of the class succeeding
the modal class $= 14.$

Now, using the formula:
$\begin{matrix} l Mode & = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h = 9 + (\frac{25 - 15}{(2 \times 25) - 15 - 14}) \times 2 = 9 + 0.48 \times 2 = 9 + 0.96 = 9.96 \end{matrix}$

Therefore, the mode weight of a cylinder is $9.96 kg .$

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Prove that:
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Q. Find the sum of the following numbers without actually adding the numbers.
(i)

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15

(ii)

1 + 3 + 5 + 7

(iii)

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17

Q. Without adding, find the sum:
(i) (1 + 3 + 5 + 7 + 9 + 11 + 13)
(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)
(iii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23)

Re-arrange suitably and find the sum in each of the following :

(i) $\frac{11}{12} + \frac{- 17}{3} + \frac{11}{2} + \frac{- 25}{2}$

(ii) $\frac{- 6}{7} + \frac{- 5}{6} + \frac{- 4}{9} + \frac{- 15}{7}$

(iii) $\frac{3}{5} + \frac{7}{3} + \frac{9}{5} + \frac{- 13}{15} + \frac{- 7}{3}$

(iv) $\frac{4}{13} + \frac{- 5}{8} + \frac{- 8}{13} + \frac{9}{13}$

(v) $\frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5}$

(vi) $\frac{1}{8} + \frac{5}{12} + \frac{2}{7} + \frac{7}{12} + \frac{9}{7} + \frac{- 5}{16}$

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Weight $(%)$	$3 - 5$	$5 - 7$	$7 - 9$	$9 - 11$	$11 - 13$	$13 - 15$
Number of gas cylinders	$16$	$13$	$15$	$25$	$14$	$17$

The following table of grouped data represents the weight (in kg) of 100 gas cylinders. Calculate the mode weight of a cylinder.Weight(%)3−55−77−99−1111−1313−15Number of gas cylinders161315251417

The following table of grouped data represents the weight (in kg) of $100$ gas cylinders. Calculate the mode weight of a cylinder.
Weight $(%)$ $3 - 5$ $5 - 7$ $7 - 9$ $9 - 11$ $11 - 13$ $13 - 15$
Number of gas cylinders $16$ $13$ $15$ $25$ $14$ $17$