Arrange 7/3, 2/5, 15/9 and 21/18 in order from largest to smallest.

Step-by-step solution

1. 1

   Choose a strategy: one common denominator for all four. Comparing numerators only works when the denominators agree, so the whole comparison reduces to finding one denominator that all of $3$, $5$, $9$ and $18$ divide.

2. 2

   Find the least common denominator. Prime-factorising: $3=3$, $5=5$, $9=3^{2}$, $18=2\cdot 3^{2}$. Taking the highest power of each prime gives

   $\text{LCD}=2\cdot 3^{2}\cdot 5=90$

3. 3

   Convert each fraction to ninetieths. Multiply top and bottom by whatever turns the denominator into $90$:

   $\frac73=\frac{7\times 30}{90}=\frac{210}{90},\qquad \frac25=\frac{2\times 18}{90}=\frac{36}{90}$

   $\frac{15}{9}=\frac{15\times 10}{90}=\frac{150}{90},\qquad \frac{21}{18}=\frac{21\times 5}{90}=\frac{105}{90}$

4. 4

   Order the numerators. With a common denominator, larger numerator means larger fraction:

   $210>150>105>36$

5. 5

   Translate back to the original fractions.

   $\boxed{\frac73>\frac{15}{9}>\frac{21}{18}>\frac25}$

6. 6

   Cross-check with decimals. $\tfrac73=2.333$, $\tfrac{15}{9}=1.667$, $\tfrac{21}{18}=1.167$, $\tfrac25=0.4$ — the same order ✓. Note $\tfrac{15}{9}$ and $\tfrac{21}{18}$ reduce to $\tfrac53$ and $\tfrac76$, which would have been a quicker route: three of the four are improper (greater than $1$) and only $\tfrac25$ is less than $1$, so it must come last.