Subjective Questions
Q1:
If mean(\(\overline X = 50\)) and \(\sum fx = 750,\) find the number of terms (N).
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Mean(\(\overline X\))= 5o<br>\(\sum fx = 750 \)<br>N = ?</p> <p>\begin{align*} Mean\:(\overline {X}) &= \frac{\sum fx}{N}\\ 50 &= \frac{750}{N}\\ or, N &= \frac{750}{50}\\ \therefore N &= 15 \: \: \: \: _{Ans} \end{align*}</p>
Q2:
If \(Mean \: (\overline{X}) = 60 \) and \(\sum fx = 960,\) find the number.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>\(Mean \: (\overline{X}) =60 \\ \sum fx = 960 \\ Number \: of \: term \: (N) = ? \)</p> <p>\begin{align*} Mean (\overline {X}) &= \frac{\sum fx}{N}\\ 60 &= \frac{960}{N}\\ or, N &= \frac{960}{60}\\ \therefore N &= 16 \: _{Ans} \end{align*}</p>
Q3:
If \(Mean \: (\overline{X}) = 12, \sum fx = 70 + 10a\) and the number of frequency (N) = 5 + a, find the value of a.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>\(Mean \: (\overline{X}) =12 \\ \sum fx = 70 +10a\\ No. \: of \: term \: (N) = 5+a \)</p> <p>\begin{align*}Mean \: (\overline{X}) &= \frac{\sum fx}{N}\\ 12 &= \frac{70 + 10a}{5 + a}\\ or, 60 +12a &=70 + 10a\\ or,12a -10a &= 70 - 60\\ or, 2a &= 10 \\ or, a &= \frac{10}{2}\\ \therefore a &= 5 \: \: _{Ans} \end{align*}</p>
Q4:
Find the value of x, If the mean of 4, 8, 12, x and 25 is 13.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>\(Mean \: (\overline{X}) = 13 \\ No. \: of \: terms \: (N) = 5 + a \)</p> <p>\begin{align*} Mean (\overline{X}) &= \frac{\sum X}{N} \\ or, 13 &= \frac{4 + 8+12+x+25}{5}\\ or, 65 &=49 + x\\ or, x &= 65 - 49 \\ x &= 16 \:\: _{Ans} \end{align*}</p>
Q5:
The mean of 4 different number is 7 and the mean of 3 different number is 12. Find the mean of 7 numbers.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>First item mean \( (\overline {X_{1}}) = 7 \)<br>Number of first item \(n_{1} = 4 \)<br>Second item mean \( (\overline {X_{2}}) = 12\)<br>No. of Second item \(n_{2}\)= 3<br>Mean of 7 items \(\overline{X}= ?\)</p> <p>\begin{align*} ( \overline {X}) &= \frac{n_1 \overline{X}_1 + n_2 \overline{X}_2}{n_1+n_2}\\ &= \frac{4 \times 7 + 3 \times 12}{4 + 3}\\ &= \frac{28 + 36}{7}\\ &= \frac{64}{7}\\ &= 9.14 \: \: \: _{ans} \end{align*}</p>
Q6:
If the mean of 10, 20, 30, 40, 50, 60 and 30 + k is 40, find the value of k.
Type: Short
Difficulty: Easy
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Answer: <p>\(Mean (\overline{X}) = 40 \\ No. \: of \: terms (N)= 7 \\ k = ? \)</p> <p>\begin{align*} Mean(\overline{X}) &= \frac{\sum X}{N} \\ or, 40 &= \frac{10+20+30+40+50+ 60+30+k}{7}\\ or, 280 &= 240 + k\\ or, k &= 280 - 240\\ \therefore k &= 40 \: \: \: \: \: \: \: \: \: _{Ans} \end{align*}</p>
Q7:
The expenditure (in rupees) of 15 days of a man is as follows. Calculate his average expenditure.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <table width="286"><tbody><tr><td>Expenditure (in Rs.)x</td> <td>Frequency(f)</td> <td>fx</td> </tr><tr><td>24</td> <td>2</td> <td>48</td> </tr><tr><td>25</td> <td>4</td> <td>100</td> </tr><tr><td>30</td> <td>3</td> <td>90</td> </tr><tr><td>35</td> <td>4</td> <td>140</td> </tr><tr><td>40</td> <td>2</td> <td>80</td> </tr></tbody></table><p>\begin{align*} Mean(\overline{X}) &= \frac{\sum fx}{N}\\ &= \frac{458}{15}\\ &= 30.53 \\ \therefore Average \: expenditure &= Rs \: 30.53 \end{align*}</p> <p></p>
Q8:
The average age of 5 students is 9 years. Out of them the ages of 4 students are 5, 7, 8 and 15 years. What was the age of the remaining students.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Let, the age of remaining students be x.</p> <p>\(average \: age (\overline{X}) = 9 \\ Number \: of \: terms (N) = 5 \)</p> <p>\begin{align*} (\overline{X}) &= \frac{\sum X}{N}\\ or, 9 &= \frac{5+7+8+15+x}{5}\\ or, 45 &= 35 + x \\ \therefore x &=45 -35 = 10 \\ \: \\ \therefore & \text{The age of remaining student is 10} \end{align*}</p>
Q9:
Construct a frequency table of class interval of 10 from the given data and find the mean.
23, 5, 17, 28, 39, 52, 21, 16, 29, 41, 25, 33, 9, 19, 34, 59, 7, 11, 51, 31, 22, 41, 32, 55, 31, 22, 41, 32, 55, 18
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="332"><tbody><tr><td>Class interval</td> <td>mid-value (m)</td> <td>frequency (f)</td> <td>fm</td> </tr><tr><td>0-10</td> <td>5</td> <td>3</td> <td>15</td> </tr><tr><td>10-20</td> <td>15</td> <td>5</td> <td>75</td> </tr><tr><td>20-30</td> <td>25</td> <td>6</td> <td>150</td> </tr><tr><td>30-40</td> <td>35</td> <td>5</td> <td>175</td> </tr><tr><td>40-50</td> <td>45</td> <td>2</td> <td>90</td> </tr><tr><td>50-60</td> <td>55</td> <td>4</td> <td>220</td> </tr><tr><td></td> <td></td> <td>N = 25</td> <td></td> </tr></tbody></table><p>We know that,</p> <p>\begin{align*} Mean (\overline{X} ) &= \frac{\sum fm}{N}\\ &= \frac{725}{25}\\ &= 29 \: _{Ans} \end{align*}</p>
Q10:
Find the mean from the following data.
Class-interval |
10-20 |
10-30 |
10-40 |
10-50 |
10-60 |
10-70 |
10-80 |
10-90 |
Frequency |
4 |
16 |
56 |
97 |
124 |
137 |
146 |
150 |
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="396"><tbody><tr><td>Class interval</td> <td>cf.</td> <td>f</td> <td>mid-value (m)</td> <td>fm</td> </tr><tr><td>10-20</td> <td>4</td> <td>4</td> <td>15</td> <td>60</td> </tr><tr><td>20-30</td> <td>16</td> <td>12</td> <td>25</td> <td>300</td> </tr><tr><td>30-40</td> <td>56</td> <td>40</td> <td>35</td> <td>1400</td> </tr><tr><td>40-50</td> <td>97</td> <td>41</td> <td>45</td> <td>1845</td> </tr><tr><td>50-60</td> <td>124</td> <td>27</td> <td>55</td> <td>1485</td> </tr><tr><td>60-70</td> <td>137</td> <td>13</td> <td>65</td> <td>845</td> </tr><tr><td>70-80</td> <td>146</td> <td>9</td> <td>75</td> <td>675</td> </tr><tr><td>80-90</td> <td>150</td> <td>4</td> <td>85</td> <td>340</td> </tr><tr><td></td> <td></td> <td>N = 150</td> <td></td> <td>\(\sum fm=6950\)</td> </tr></tbody></table><p>\begin{align*}Mean \: (\overline{X})&= \frac{\sum fm}{N}\\ &= \frac{6950}{150}\\ &= 46.33 \: _{Ans} \end{align*}</p>
Q11:
Calculate the average wages from the following data.
Wages |
50-60 |
50-70 |
50-80 |
50-90 |
50-100 |
No. of people |
6 |
14 |
26 |
34 |
40 |
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="360"><tbody><tr><td>Wages</td> <td>cf</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>50-60</td> <td>6</td> <td>6</td> <td>55</td> <td>330</td> </tr><tr><td>60-70</td> <td>14</td> <td>8</td> <td>65</td> <td>520</td> </tr><tr><td>70-80</td> <td>26</td> <td>12</td> <td>75</td> <td>900</td> </tr><tr><td>80-90</td> <td>34</td> <td>8</td> <td>85</td> <td>680</td> </tr><tr><td>90-100</td> <td>40</td> <td>6</td> <td>95</td> <td>570</td> </tr><tr><td></td> <td></td> <td>N=40</td> <td></td> <td>\(\sum fm= 3000\)</td> </tr></tbody></table><p>\begin{align*} Mean (\overline {X})&= \frac{\sum fm}{N}\\ &= \frac{3000}{40}\\ &= 75 \: _{Ans} \end{align*}</p>
Q12:
Calculate the average wages from the following data.
Wages |
50-60 |
50-70 |
50-80 |
50-90 |
50-100 |
No. of people |
6 |
14 |
26 |
34 |
40 |
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="360"><tbody><tr><td>Wages</td> <td>cf</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>50-60</td> <td>6</td> <td>6</td> <td>55</td> <td>330</td> </tr><tr><td>60-70</td> <td>14</td> <td>8</td> <td>65</td> <td>520</td> </tr><tr><td>70-80</td> <td>26</td> <td>12</td> <td>75</td> <td>900</td> </tr><tr><td>80-90</td> <td>34</td> <td>8</td> <td>85</td> <td>680</td> </tr><tr><td>90-100</td> <td>40</td> <td>6</td> <td>95</td> <td>570</td> </tr><tr><td></td> <td></td> <td>N=40</td> <td></td> <td>\(\sum fm= 3000\)</td> </tr></tbody></table><p>\begin{align*} Mean (\overline {X})&= \frac{\sum fm}{N}\\ &= \frac{3000}{40}\\ &= 75 \: _{Ans} \end{align*}</p>
Q13:
Calculate the mean from the following data.
X |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
f |
5 |
7 |
8 |
4 |
6 |
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="337"><tbody><tr><td>class interval</td> <td>midvalue</td> <td>f</td> <td>fm</td> </tr><tr><td>0-10</td> <td>5</td> <td>5</td> <td>25</td> </tr><tr><td>10-20</td> <td>15</td> <td>7</td> <td>105</td> </tr><tr><td>20-30</td> <td>25</td> <td>8</td> <td>200</td> </tr><tr><td>30-40</td> <td>35</td> <td>4</td> <td>140</td> </tr><tr><td>40-50</td> <td>45</td> <td>6</td> <td>270</td> </tr><tr><td></td> <td></td> <td>N= 30</td> <td>\( \sum fm = 740 \)</td> </tr></tbody></table><p>\begin{align*} Mean (\overline{X}) &= \frac{\sum fm}{N}\\ &= \frac{740}{30}\\ &= 24.6 \: _{ans} \end{align*}</p>
Q14:
Compute the mean from the table given below.
Marks |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
No. of students |
2 |
5 |
7 |
6 |
3 |
2 |
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>Calculation of mean</p> <table width="227"><tbody><tr><td>Marks</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>10-20</td> <td>2</td> <td>15</td> <td>30</td> </tr><tr><td>20-30</td> <td>5</td> <td>25</td> <td>125</td> </tr><tr><td>30-40</td> <td>7</td> <td>35</td> <td>245</td> </tr><tr><td>40-50</td> <td>6</td> <td>45</td> <td>270</td> </tr><tr><td>50-60</td> <td>3</td> <td>55</td> <td>165</td> </tr><tr><td>60-70</td> <td>2</td> <td>65</td> <td>130</td> </tr><tr><td></td> <td>N=25</td> <td></td> <td>\(\sum fm = 965\)</td> </tr></tbody></table><p>We know that,</p> <p>\begin{align*} Mean (\overline {X}) &= \frac{\sum fm }{N} \\ &= \frac{965}{25}\\ &= 38.6 \: \: _{Ans} \end{align*}</p> <p></p> <p></p> <p></p>
Q15:
If the mean is 32, find the value of p from the following.
x |
5-15 |
15-25 |
25-35 |
35-45 |
45-55 |
55-65 |
f |
5 |
8 |
p |
9 |
7 |
1 |
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>To find the value of p.</p> <table width="389"><tbody><tr><td>X</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>5-15</td> <td>5</td> <td>10</td> <td>50</td> </tr><tr><td>15-25</td> <td>8</td> <td>20</td> <td>160</td> </tr><tr><td>25-35</td> <td>p</td> <td>30</td> <td>30p</td> </tr><tr><td>35-45</td> <td>9</td> <td>40</td> <td>360</td> </tr><tr><td>45-55</td> <td>7</td> <td>50</td> <td>350</td> </tr><tr><td>55-65</td> <td>1</td> <td>60</td> <td>60</td> </tr><tr><td></td> <td>N = p + 30</td> <td></td> <td>\( \sum fm = 30p+ 980 \)</td> </tr></tbody></table><p>We know that,</p> <p>\begin{align*}Mean (\overline{X})&= \frac{\sum fm}{N}\\ 32 &= \frac{30p + 980}{p + 30}\\ or, 32p + 960 &= 30p + 980\\ or, 32p -30p &= 980 - 960 \\ or, p &= \frac{20}{2}\\ \therefore p&=10 \: _{Ans} \end{align*}</p>
Q16:
The mean of the data given below is 36, find the value of p.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>To find the value of p.</p> <p><strong>Solution:</strong></p> <p>To find the value of p.</p> <table width="389"><tbody><tr><td>Age in year</td> <td>No. of teachers (f)</td> <td>m</td> <td>fm</td> </tr><tr><td>10-20</td> <td>3</td> <td>15</td> <td>45</td> </tr><tr><td>20-30</td> <td>8</td> <td>25</td> <td>200</td> </tr><tr><td>30-40</td> <td>15</td> <td>35</td> <td>525</td> </tr><tr><td>40-50</td> <td>p</td> <td>45</td> <td>45p</td> </tr><tr><td>50-60</td> <td>4</td> <td>55</td> <td>220</td> </tr><tr><td></td> <td>N=p+30</td> <td></td> <td>\( \sum fm = 45p + 990\)</td> </tr></tbody></table><p>We know that,</p> <p>\begin{align*} Mean \: (\overline {X}) &= \frac{\sum fm}{N}\\ 36 &= \frac{45p + 990}{p+30}\\ or, 1080 + 36p &= 45p +990\\ or, 45p - 36p &= 1080 -990 \\ or, 9p &= 90 \\ \therefore p &= \frac{90}{9} = 10 \: _{Ans} \end{align*}</p>
Q17:
The mean of the data given below is 36, find the value of p.
Type: Short
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>To find the value of p.</p> <p><strong>Solution:</strong></p> <p>To find the value of p.</p> <table width="389"><tbody><tr><td>Age in year</td> <td>No. of teachers (f)</td> <td>m</td> <td>fm</td> </tr><tr><td>10-20</td> <td>3</td> <td>15</td> <td>45</td> </tr><tr><td>20-30</td> <td>8</td> <td>25</td> <td>200</td> </tr><tr><td>30-40</td> <td>15</td> <td>35</td> <td>525</td> </tr><tr><td>40-50</td> <td>p</td> <td>45</td> <td>45p</td> </tr><tr><td>50-60</td> <td>4</td> <td>55</td> <td>220</td> </tr><tr><td></td> <td>N=p+30</td> <td></td> <td>\( \sum fm = 45p + 990\)</td> </tr></tbody></table><p>We know that,</p> <p>\begin{align*} Mean \: (\overline {X}) &= \frac{\sum fm}{N}\\ 36 &= \frac{45p + 990}{p+30}\\ or, 1080 + 36p &= 45p +990\\ or, 45p - 36p &= 1080 -990 \\ or, 9p &= 90 \\ \therefore p &= \frac{90}{9} = 10 \: _{Ans} \end{align*}</p>
Q18:
If the mean of the data given below is 39. Calculate the value of m.
Wages |
15-25 |
25-35 |
35-45 |
45-55 |
55-65 |
No. of workers |
4 |
6 |
12 |
m |
3 |
Type: Long
Difficulty: Easy
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Answer: <p><strong>Solution:</strong></p> <p>To find the value of m.</p> <table width="346"><tbody><tr><td>Wages</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>15-25</td> <td>4</td> <td>20</td> <td>80</td> </tr><tr><td>25-35</td> <td>6</td> <td>30</td> <td>180</td> </tr><tr><td>35-45</td> <td>12</td> <td>40</td> <td>480</td> </tr><tr><td>45-55</td> <td>m</td> <td>50</td> <td>50m</td> </tr><tr><td>55-65</td> <td>3</td> <td>60</td> <td>180</td> </tr><tr><td></td> <td>N = m + 25</td> <td></td> <td>\( \sum fm = 50m + 920\)</td> </tr></tbody></table><p>\begin{align*} Mean (\overline{X}) &= \frac{\sum fm}{N}\\ 39 &= \frac{50m+920}{m+25}\\ or, 39m + 975 &= 50m + 920 \\ or, 50m - 39m &= 975 - 920 \\ or, 11m &= 55 \\ or, x&=\frac{55}{11}\\ \therefore m &= 5 \: _{Ans} \end{align*}</p> <p></p>
Q19:
If the mean of the following data is 53, find the value k.
Class |
0-20 |
20-40 |
40-60 |
60-80 |
80-100 |
Frequency |
15 |
k |
21 |
29 |
17 |
Type: Long
Difficulty: Easy
Show/Hide Answer
Answer: <p><strong>Solution:</strong></p> <p>To find the value of k.</p> <table width="245"><tbody><tr><td>Class</td> <td>f</td> <td>m</td> <td>fm</td> </tr><tr><td>0-20</td> <td>15</td> <td>10</td> <td>150</td> </tr><tr><td>20-40</td> <td>k</td> <td>30</td> <td>30k</td> </tr><tr><td>40-60</td> <td>21</td> <td>50</td> <td>1050</td> </tr><tr><td>60-80</td> <td>29</td> <td>70</td> <td>2030</td> </tr><tr><td>80-100</td> <td>17</td> <td>90</td> <td>1530</td> </tr><tr><td></td> <td>N=82+k</td> <td></td> <td>\( \sum fm = 30k + 4760 \)</td> </tr></tbody></table><p></p> <p>\begin{align*} Mean \: (\overline{X} ) &= \frac{\sum fm}{N}\\ 53 &= \frac{30k + 4760}{82 + k}\\ or, 4346 + 53k &= 30k + 4760 \\ or, 53k - 30k &= 4760 - 4346 \\ or, 23k &= 414 \\ or, k &= \frac{414}{23}\\ \therefore k &= 18 \: _{Ans} \end{align*}</p> <p></p>
Q20:
Calculate mean from the following data.
Marks |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
Frequency |
12 |
7 |
8 |
3 |
10 |
Type: Long
Difficulty: Easy
Show/Hide Answer
Answer: <p><strong>Solution:</strong></p> <p>Calculating mean</p> <table width="395"><tbody><tr><td>Marks</td> <td>frequency (f)</td> <td>mid value (m)</td> <td>fm</td> </tr><tr><td>20-30</td> <td>12</td> <td>25</td> <td>300</td> </tr><tr><td>30-40</td> <td>7</td> <td>35</td> <td>245</td> </tr><tr><td>40-50</td> <td>8</td> <td>45</td> <td>360</td> </tr><tr><td>50-60</td> <td>3</td> <td>55</td> <td>165</td> </tr><tr><td>60-70</td> <td>10</td> <td>65</td> <td>650</td> </tr><tr><td></td> <td>N = 40</td> <td></td> <td>\(\sum fm = 1720\)</td> </tr></tbody></table><p>\begin{align*} Mean \: (\overline{X}) &= \frac{\sum fm}{N}\\ &= \frac{1720}{40}\\ &= 43 \: _{Ans} \end{align*}</p>
Q21:
The mean and total number of students are 31 and 50 find the missing frequency.
Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
No. of students |
4 |
x |
10 |
y |
6 |
4 |
Type: Long
Difficulty: Easy
Show/Hide Answer
Answer: <p><strong>Solution:</strong></p> <p>To find the value of x and y</p> <table width="320"><tbody><tr><td>Marks</td> <td>f</td> <td>mid value</td> <td>fm</td> </tr><tr><td>0-10</td> <td>4</td> <td>5</td> <td>20 </td> </tr><tr><td>10-20</td> <td>x</td> <td>15</td> <td>15x</td> </tr><tr><td>20-30</td> <td>10</td> <td>25 </td> <td>250</td> </tr><tr><td>30-40</td> <td>y</td> <td>35</td> <td>35y</td> </tr><tr><td>40-50</td> <td>6</td> <td>45</td> <td>270</td> </tr><tr><td>50-60</td> <td>4</td> <td>55 </td> <td>220</td> </tr><tr><td></td> <td>\( N= x+y +24\)</td> <td></td> <td>\(\sum fm= 15x +35y +760 \)</td> </tr></tbody></table><p>\begin{align*} Mean (\overline{X}) &= \frac{\sum fm}{N}\\ 31 &= \frac{15x +35y+760}{50}\\ or, 1150 - 760 &= 15x+35y \\ or, 15x + 35y &= 790 \: ... .. .. . .. (1) \\ x+y+24 &= 50 \\ or, y&=50-24-x \\ y&=26-x ....................(2) \\ Putting \: valu&e \: of \: y \: in\:equation \: ....(1)\\ 15x + 35y&=790\\ or, 15x + 35(26-x) &= 790 \\ or, 15x+910-35x &= 790\\ or, -20x &= 790-910 \\ or, x &= \frac{-120}{-20}\\ \therefore x &= 6 \\ \: \\ putting \: value \: of \: x & \: in \: equation .. (2) \\ y &= 26-6 \\ \therefore y&=20 \end{align*}</p>