In this chapter we will learn to arrange two or more rational number in ascending order.
In ascending order, we arrange the number from smallest to greatest.
Method to arrange number in ascending order
To arrange the numbers, we first have to compare the given rational numbers and then arrange as per the value from smallest to greatest.
To arrange the numbers, follow the below steps;
(a) write all the number in fraction form
(b) Find LCM of all denominators
(c) Multiply the rational number to make denominator equal to LCM
(d) Now simply compare the numerator and arrange the number
I hope you understood the above process. Let us look at the solved examples for better clarity.
Example 01
Arrange rational numbers in ascending orders \mathtt{\frac{7}{4} \ ,\ \frac{2}{5} \ \&\ \frac{\ 6}{7}}
Solution
Follow the below steps;
(a) Take LCM of denominator
LCM (4, 5, 7) = 140
(b) Multiply numerator and denominator of all rational number such that their denominator becomes 140.
(i) Rational number 7/4
Multiply numerator and denominator by 35
\mathtt{\Longrightarrow \frac{7}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 35}{4\times 35}}\\\ \\ \mathtt{\Longrightarrow \ \frac{245}{140}}
(ii) Rational number 2 / 5
Multiply numerator and denominator by 28.
\mathtt{\Longrightarrow \frac{2}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 28}{5\times 28}}\\\ \\ \mathtt{\Longrightarrow \ \frac{56}{140}}
(iii) Rational number 6 / 7
Multiply numerator and denominator by 20
\mathtt{\Longrightarrow \frac{6}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\times 20}{7\times 20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{120}{140}}
Now we have got the rational number with same denominator.
(c) Compare the numerator of rational numbers and arrange it from smallest to greatest.
\mathtt{\Longrightarrow \frac{56}{140} < \ \frac{120}{140} < \frac{245}{140}}
Putting the original values;
\mathtt{\Longrightarrow \frac{2}{5} < \ \frac{6}{7} < \frac{7}{4}}
Example 02
Arrange the rational numbers in ascending order \mathtt{\frac{10}{3} \ ,\ \frac{16}{9} \ \&\ \frac{\ 1}{4}}
Solution
Do the following steps;
(a) Find LCM of denominators
LCM (3, 9, 4) = 36
(b) Multiply the fractions to make denominator equal to 36
(i) Rational number 10 / 3
Multiply numerator and denominator by 12.
\mathtt{\Longrightarrow \frac{10}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 12}{3\times 12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{120}{36}}
(ii) Rational number 16 / 9
Multiply numerator and denominator by 4
\mathtt{\Longrightarrow \frac{16}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{16\times 4}{9\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{64}{36}}
(iii) Rational number 1/ 4
Multiply numerator and denominator by 9
\mathtt{\Longrightarrow \frac{1}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 9}{4\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{36}}
Now we have rational numbers with same denominators so comparison can easily be done.
(c) Compare the numerators and arrange in ascending orders.
\mathtt{\Longrightarrow \frac{9}{36} < \ \frac{64}{36} < \frac{120}{36}}
Putting the original values, we get;
\mathtt{\Longrightarrow \frac{1}{4} < \ \frac{16}{9} < \frac{10}{3}}
Example 03
Given are rational numbers \mathtt{\frac{13}{5} ,\ \frac{-7}{2} \ ,\ \frac{-15}{4} \ \&\ \frac{\ 17}{9}}
Arrange the rational numbers in ascending orders
Solution
Note that among the given numbers, two are negative & two are positive.
We know that negative numbers are always less than positive numbers.
So we will do two comparisons separately. One between the negative numbers and other between positive numbers.
Comparing -7 / 2 and -15 / 4
Follow the below steps;
(a) Find LCM of denominator.
LCM (2, 4) = 4
(b) Multiply fractions to make denominator 4.
(i) Fraction -7 / 2
Multiply numerator and denominator by 2.
\mathtt{\Longrightarrow \frac{-7}{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-7\times 2}{2\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-14}{4}}
(ii) Fraction -15 / 4
Denominator is already 4.
Comparing the two numbers;
\mathtt{\Longrightarrow \frac{-14}{4} >\ \frac{-15}{4}}
Putting the original values;
\mathtt{\Longrightarrow \frac{-7}{2} >\ \frac{-15}{4}}
Now let’s compare the positive numbers \mathtt{\frac{13}{5} \ \&\ \frac{\ 17}{9}}
(a) Find LCM of denominators
LCM (5, 9) = 45
(b) Multiply the rational numbers to make denominator 45
(i) Number 13 / 5
Multiply numerators and denominators by 9
\mathtt{\Longrightarrow \frac{13}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{13\times 9}{5\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{117}{45}}
(ii) Number 17 / 9
Multiply numerator and denominator by 5
\mathtt{\Longrightarrow \frac{17}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{17\times 5}{9\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{85}{45}}
Now we got the rational numbers with same denominator.
(c) Compare the numerators.
\mathtt{\Longrightarrow \frac{117}{45} >\ \frac{85}{45}}
Putting the original values;
\mathtt{\Longrightarrow \frac{13}{5} >\ \frac{17}{9}}
Combining all the result, the final solution will be;
\mathtt{\Longrightarrow \frac{-15}{4} < \ \frac{-7}{2} < \frac{17}{9} < \frac{13}{5}}
Example 04
Arrange the below rational number in ascending order.
\mathtt{\Longrightarrow \frac{3}{16} ,\ \frac{1}{9} \ \&\ \frac{3}{8} \ }
Solution
Do the following steps;
(a) Find LCM of denominators.
LCM (16, 9, 8) = 144
(b) Multiply each rational number to make denominator 144
(i) Rational number 3/16
Multiply numerator and denominator by 9
\mathtt{\Longrightarrow \frac{3}{16}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 9}{16\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{27}{144}}
(ii) Number 1/9
Multiply numerator and denominator by 16
\mathtt{\Longrightarrow \frac{1}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 16}{9\times 16}}\\\ \\ \mathtt{\Longrightarrow \ \frac{16}{144}}
(iii) Number 3 / 8
Multiply numerator and denominator by 18
\mathtt{\Longrightarrow \frac{3}{8}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 18}{8\times 18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{54}{144}}
We got the rational numbers with same denominator. Now simply compare the numerators.
(c) Comparing the numerators we get;
\mathtt{\Longrightarrow \frac{16}{144} < \ \frac{27}{144} < \frac{54}{144}}
Putting the original values;
\mathtt{\Longrightarrow \frac{1}{9} < \ \frac{3}{16} < \frac{3}{8}}