Question 12 Marks
Find the mean proportional between: $0.4$ and $0.9$
Answer$0.4 : \text{x} :: \text{x} : 0.9$
$\Rightarrow \text{x}^2 = 0.4 \times 0.9$
$\text{x} = \sqrt{0.36} = 0.6$
Mean proportional $= 0.6$
View full question & answer→Question 22 Marks
Find the third proportional to: $4.5$ and $6$
AnswerLet $x$ be the third proportional, then $4.5 : 6 : : 6 : \text{x}$
$\Rightarrow4.5:6::6:\text{x}$
$\Rightarrow4.5\times\text{x}=6\times6$
$\Rightarrow\text{x}=\frac{6\times6}{4.5}=\frac{6\times6\times10}{45}=\frac{360}{45}=8$
$\therefore$ Third proportional $= 8$
View full question & answer→Question 32 Marks
Find the fourth proportional to the number: $8, 36, 6$
AnswerLet $x$ be the fourth proportional,
then: $8:36::6:\text{x}$
$\therefore8\times\text{x}=36\times6$
$\Rightarrow\text{x}=\frac{36\times6}{8}$
$\text{x}=9\times3=27$
$\therefore$ Fourth proportional $= 27$
View full question & answer→Question 42 Marks
Express the following ratios in the simplest form: $8$ months $: 1$ year
Answer$8$ months $: 1$
year $= 8$ months $: 12$ months $($Converting into the same kind$) HCF$ of $8$ and $12 = 4$
Dividing by $4 8 ÷ 4 : 12 ÷ 4 = 2 : 3$
View full question & answer→Question 52 Marks
Two numbers are in the ratio $5 : 7.$ If the sum of these numbers is $84,$ find the numbers.
AnswerSuppose that the numbers are $5x$ and $7x$
Then, $5x + 7x = 84$
$12x = 84$
$x = 7$
Hence, the numbers are $(5 \times 7) = 35$ and $(7 \times 7) = 49.$
View full question & answer→Question 62 Marks
Find the fourth proportional to the number:
$5, 7, 30$
AnswerLet $x$ be the fourth proportional, then:
$5:7::30:\text{x}$
$\therefore5\times\text{x}=7\times30$
$\Rightarrow\text{x}=\frac{7\times30}{5}=7\times6$
$\text{x}=42$
$\therefore$ Fourth proportional $= 42$
View full question & answer→Question 72 Marks
If a car covers $54\ km$ in an hour, how much distance will it cover in $40$ minutes$ ?$
AnswerDistance covered in $60$ min $= 54\ km$
Distance covered in $1$ min $=\frac{54}{60}\text{km}$
Distance covered in $40$ min $=-\frac{54}{60}\times40=36\text{km}$
View full question & answer→Question 82 Marks
Find the fourth proportional to the number:
$2, 8, 14, 3.5$
AnswerLet $x$ be the fourth proportional, then:
$2.8 : 14 ::3.5 : \text{x}$
$\therefore2.8\times\text{x}=14\times3.5$
$\Rightarrow\text{x}=\frac{14\times3.5}{2.8}=\frac{14\times5}{4}=\frac{7\times5}{2}$
$\text{x}=\frac{35}{2}=17.5$
$\therefore$ Fourth proportional $= 17.5$
View full question & answer→Question 92 Marks
Find the mean proportional between: $3$ and $27$
Answer$3 : \text{x} : : \text{x} : 27$
$\Rightarrow\text{x}^2 = 3 \times 27 = 81$
$\text{x} = \sqrt{81} = 9$
Mean proportional $= 9$
View full question & answer→Question 102 Marks
Find the mean proportional between: $6$ and $24$
AnswerLet $x$ be the mean proportional,
then $6:\text{x}::\text{x}:24$
$\Rightarrow\text{x}^2=6\times24=144$
$\text{x}=\sqrt{144}=12$
Mean proportional $= 12$
View full question & answer→Question 112 Marks
Express the following ratios in the simplest form: $(2\ kg\ 250\ g) : (3\ kg)$
Answer$2\ kg\ 250\ g : 3\ kg = 2250\ g : 3000\ g ($converting into the same kind$)$
$HCF$ of $2250$ and $3000 = 750$
Dividing by $750, 2250 ÷ 750 : 3000 ÷ 750 = 3 : 4$
View full question & answer→Question 122 Marks
If $8 : x :: 16 : 35,$ find the value of $x.$
AnswerProduct of the extremes $= 8 \times 35 = 280$
Product of the means $= 16 \times x = 16x$
Since $8 : x :: 16 : 35$
We have: Product of the extremes $=$ product of the means
$\Rightarrow 280 = 16x$
$\Rightarrow x = 17.5$
View full question & answer→Question 132 Marks
If $(4x + 5) : (3x + 11) = 13 : 17,$ find the value of $x.$
Answer$(4x + 5) : (3x + 11) = 13 : 17$
$\frac{4\text{x}+5}{3\text{x}+11}=\frac{13}{17}$
By cross multiplication, $68x + 85 = 39x + 143$
$\Rightarrow 68x - 39x = 143 - 85$
$\Rightarrow 29x = 58$
$x = 2$
Hence $x = 2$
View full question & answer→Question 142 Marks
Show that $30, 40, 45, 60$ are in proportion.
AnswerWe know that $a, b, c, d$ are in proportion if $ad = bc$
Now $30, 40, 45, 60$ are in proportion if $30 \times 60 = 40 \times 45$
if $1800 = 1800$ which is true $30, 40, 45, 60$ are in proportion.
View full question & answer→Question 152 Marks
Express the following ratios in the simplest form: $1\ m\ 5\ cm : 63\ cm$
Answer$1\ m\ 5\ cm : 63\ cm = 105\ cm : 63\ cm ($Converting into same kind$) HCF$ of $105$ and $63 = 21$
$105 ÷ 21 : 63 ÷ 21 ($Dividing by $21) = 5 : 3$
View full question & answer→Question 162 Marks
Find the third proportional to: $8$ and $12$
AnswerLet $x$ be the third proportional,
then $8 : 12 : : 12 : \text{x}$
$\Rightarrow8\times\text{x}=12\times12$
$\Rightarrow\text{x}=\frac{12\times12}{8}=18$
$\therefore$ Third proportional $= 18$
View full question & answer→Question 172 Marks
The ratio of monthly income to the savings of a family is $11 : 2.$ If the savings be $Rs. 2500,$ find the income and expenditure.
AnswerRatio in income and savings $= 11 : 2$
Let income $= 11x$
Then savings $= 2x$
But savings $= Rs. 2500$
$2x = 2500$
$\Rightarrow x = 1250$
Then income $= 1250 \times 11 = Rs. 13750$ and e
xpenditure $=$ Total income $–$ savings
$= 13750 - 2500 = Rs. 11250$
View full question & answer→Question 182 Marks
Express the following ratios in the simplest form: $2.5 : 6.5 : 8$
Answer$2.5 : 6.5 : 8$
$= 2.5 : 6.5 : 8.0$
$= 25 : 65 : 80$
$HCF$ of $25, 65, 80 = 5$
Dividing by $5, 5 : 13 : 16$
View full question & answer→Question 192 Marks
If $x, 18, 108$ are in continued proportion, find the value of $x.$
Answer$x : 18 :: 18 : 108$
$\Rightarrow x \times 108 = 18 \times 18 ($Product of extremes $=$ Product of means$)$
$\Rightarrow 108x = 324$
$\Rightarrow x = 3$
Hence, the value of $x$ is $3.$
View full question & answer→Question 202 Marks
If $40$ men can finish a piece of work in $60$ days, in how many days will $75$ men finish the same work$?$
Answer$40$ men can finish the work in $60$ days
$1$ man can finish the work in $60 \times 40$ days $[$Less men, more days$]$
$75$ men will finish the work in $=\frac{60\times40}{75}=32$ days
Hence, $75$ men will finish the same work in $32$ days.
View full question & answer→Question 212 Marks
If the third proportional to $7$ and $x$ is $28,$ find the value of $x.$
Hint. $7 : x :: x : 28$ Find $x.$
Answer$\text { Third proportional }=28 \text { then }$
$7: x:: x: 28$
$\Rightarrow 7 \times 28=x \times x$
$\Rightarrow x^2=28 \times 7=196$
$\Rightarrow x=\sqrt{ } 196=14$
$x=14$
View full question & answer→Question 222 Marks
Express the following ratios in the simplest form: $75$ paise $: 3$ rupees
Answer$75$ paise $: 3$ rupees $= 75$ paise $: 300$ paise $($Converting into same kind$)$
$HCF$ of $75, 300 = 75$
$75 : 300 = 75 ÷ 75 : 300 ÷ 75 ($Dividing by $75) = 1 : 4$
View full question & answer→Question 232 Marks
Show that $36, 49, 6, 7$ are not in proportion.
AnswerWe know that if $a, b, c, d$ are in proportion if $ad = bc$
Now $36, 49, 6, 7$ are in proportion
if $36 \times 7 = 49 \times 6$
if $252 = 294$
But $252 \neq 294$
$36, 49, 6, 7$ are not in proportion.
View full question & answer→Question 242 Marks
Express the following ratios in the simplest form: $1\ km : 750\ m$
Answer$1\ km : 750\ m = 1000\ m : 750\ m ($Converting into metre$) = 4 : 3 ($dividing by $250) = 4 : 3$
View full question & answer→Question 252 Marks
The ratio of boys and girls in a school is $8 : 3.$ If the total number of girls be $375,$ find the number of boys in the school.
AnswerRatio in boys and girls $= 8 : 3$ And
total number of girls $= 375$
Let number of boys $= 8x$
Then number of girls$ = 3x$
$3x = 375$
$\Rightarrow x = 125$
Number of boys$ = 8x$
$= 8 \times 125 = 1000$
View full question & answer→Question 262 Marks
Find the third proportional to $8$ and $12.$
AnswerSuppose that the third proportional to $8$ and $12$ is $x$
Then, $8 : 12 :: 12 : x $
$\Rightarrow 8x = 144 ($Product of extremes $=$ Product of means$) $
$x = 18$
Hence, the third proportional is $18.$
View full question & answer→Question 272 Marks
If $x : 35 :: 48 : 60$ find the value of $x.$
AnswerProduct of the extremes $= x \times 60 = 60x$
Product of the means $= 35 \times 48 = 1680$
Since $x : 35 :: 48 : 60$
We have: Product of the extremes $=$ product of the means
$\Rightarrow 60x = 1680 $
$\Rightarrow x = 28$
View full question & answer→Question 282 Marks
Find the third proportional to: $12$ and $18$
AnswerLet $x$ be the third proportional,
then $12 : 18 : : 18 : \text{x}$
$\Rightarrow12\times\text{x}=18\times18$
$\Rightarrow\text{x}=\frac{18\times18}{12}=\frac{3\times18}{2}=3\times9=27$
$\therefore$ Third proportional $= 27$
View full question & answer→Question 292 Marks
Express the following ratios in the simplest form: $1$ hour $5$ minutes $: 45$ minutes
Answer$1$ hour $5$ minutes $: 45$ minutes
$= 65$ minutes $: 45$ minutes $($Converting into minutes$)$
$13 : 9 ($dividing by $5) = 13 : 9$
View full question & answer→Question 302 Marks
Show that the numbers $25, 36, 5, 6$ are not in proportion.
AnswerProduct of the extremes $= 25 \times 6 = 150$
Product of the means $= 36 \times 5 = 180$
The product of the extremes is not equal to that of the means.
Hence, $25, 36, 5$ and $6$ are not in proportion.
View full question & answer→