Exercises — Equivalent fractions, simplifying fractions

A fraction equals $\frac{1}{2}$ when the bottom is exactly twice the top (because $\frac{1}{2}=\frac{1\times k}{2\times k}$, so bottom $=2\times$ top).

• $\frac{3}{6}$: $6=2\times 3$ ✓
• $\frac{4}{9}$: $9\ne 2\times 4=8$ ✗
• $\frac{5}{10}$: $10=2\times 5$ ✓
• $\frac{6}{10}$: $10\ne 2\times 6=12$ ✗

Answer: $\frac{3}{6}$ and $\frac{5}{10}$.

The bottom went $5\to 20$, which is $\times 4$ (since $5\times 4=20$). By the rule we must do the same to the top: $2\times 4=8$. Answer: $\frac{2}{5}=\frac{8}{20}$.

Method 1 (divide by GCD). $\gcd (24,36)=12$ (biggest number dividing both). $\frac{24÷12}{36÷12}=\frac{2}{3}$. Method 2 (peel factors). Both even: $\frac{24}{36}\to \frac{12}{18}\to \frac{6}{9}$. Now both divide by $3$: $\frac{6}{9}\to \frac{2}{3}$. Since $\gcd (2,3)=1$, stop. Answer: $\frac{2}{3}$.

Find $k$: we need $7\times k=63\Rightarrow k=9$. Multiply top too: $4\times 9=36$. Answer: $\frac{4}{7}=\frac{36}{63}$.

Cross-multiply: $14\times 33=462$ and $21\times 22=462$. Equal, so yes. (Both simplify to $\frac{2}{3}$: $\frac{14}{21}÷7=\frac{2}{3}$, $\frac{22}{33}÷11=\frac{2}{3}$.)

Simplify $\frac{35}{56}$. Prime-factorise (see Prime factorisation): $35=5\times 7$, $56=2^3\times 7$. The shared factor is $7$, so $\gcd =7$. $\frac{35÷7}{56÷7}=\frac{5}{8}$. Now $\gcd (5,8)=1$ ✓. Answer: $a=5,b=8$.

$\frac{n}{12}$ is lowest terms when $\gcd (n,12)=1$, i.e. $n$ shares no factor with $12=2^2\times 3$. So $n$ must be not a multiple of $2$ and not a multiple of $3$. Check $1$–$10$: $1$ ✓, $2$ ✗, $3$ ✗, $4$ ✗, $5$ ✓, $6$ ✗, $7$ ✓, $8$ ✗, $9$ ✗, $10$ ✗. Answer: $n=1,5,7$.

$\frac{18}{30}$: $\gcd (18,30)=6\Rightarrow \frac{3}{5}$. $\frac{27}{45}$: $\gcd (27,45)=9\Rightarrow \frac{3}{5}$. Both are $\frac{3}{5}$ — the shared lowest-terms identity. (Cross-check: $18\times 45=810=30\times 27$ ✓.)

Simplify the ratio (this is the same machinery as Ratios and proportion): $\gcd (150,240)=30$, so $\frac{150÷30}{240÷30}=\frac{5}{8}$. Meaning: $5$ parts flour to $8$ parts sugar. Set up an equivalent fraction with flour $=50$: $\frac{5}{8}=\frac{50}{\square }.$ Top went $5\to 50$, that's $\times 10$, so bottom $=8\times 10=80$. Answer: simplest ratio $\frac{5}{8}$; with $50$ g flour you need $\boxed{80\text{ g sugar}}$.

Same denominator, so add the tops (see Adding and subtracting fractions): $\frac{35+15}{100}=\frac{50}{100}$. Simplify: $\gcd (50,100)=50\Rightarrow \frac{50}{100}=\frac{1}{2}$. As a decimal: $\frac{1}{2}=0.5$. As a percent: $0.5=50\%$. Answer: $\frac{1}{2}=0.5=50\%$.

$20\times 5=100$, so multiply top too: $7\times 5=35$. Thus $\frac{7}{20}=\frac{35}{100}$. "Out of $100$" is percent, so $\frac{35}{100}=35\%$. Answer: $\frac{35}{100}=35\%$.

Equal fractions ⇒ cross-multiply: $a\times 5=2\times (a+12)$. $5a=2a+24\Rightarrow 3a=24\Rightarrow a=8.$ Check: $\frac{8}{8+12}=\frac{8}{20}=\frac{2}{5}$ ✓. Answer: $a=8$.

Since $\frac{a}{b}=\frac{3}{7}$, write $a=3k$, $b=7k$ for some $k$ (every equivalent fraction is $\frac{3}{7}$ scaled by $k$). Their sum: $3k+7k=10k=40\Rightarrow k=4$. So $a=3\times 4=12$, $b=7\times 4=28$. Answer: $\frac{12}{28}$ (which is indeed $\frac{3}{7}$; $12+28=40$ ✓).

Suppose some whole number $d$ divides both $n$ and $n+1$. Then $d$ must also divide their difference: $(n+1)-n=1.$ The only whole number that divides $1$ is $d=1$. So the biggest common factor of $n$ and $n+1$ is $1$, i.e. $\gcd (n,n+1)=1$. Therefore $\frac{n}{n+1}$ can never be simplified — it is always already in lowest terms. ∎ (Try it: $\frac{7}{8},\frac{99}{100},\frac{1000}{1001}$ — none simplify.)

Simplify $\frac{24}{36}$. ::: $\frac{2}{3}$ Write $\frac{4}{7}$ with denominator $63$. ::: $\frac{36}{63}$ $\frac{a}{a+12}=\frac{2}{5}$, find $a$. ::: $a=8$ Why is $\frac{n}{n+1}$ always lowest terms? ::: Any common divisor must divide $(n+1)-n=1$, so $\gcd =1$. $\frac{7}{20}$ as a percent? ::: $35\%$