Ratios and Rates

Boys: $30 - 12 = 18$. So boys : girls = 18 : 12. Divide both by the HCF, 6: $18 : 12 = 3 : 2$. (2:3 reverses the order; 18:30 is boys:total.)

Compare the rates, not the raw distances: the car does $\dfrac{100}{2} = 50$ km/h and the truck does $\dfrac{120}{3} = 40$ km/h. The car is faster. (Judging by total distance alone ignores the different times.)

A ratio of $2 : 3$ means for every 2 cars there are 3 motorbikes, so motorbikes outnumber cars. It gives the proportion, not the actual counts (there could be 4 cars and 6 motorbikes, etc.).

Convert to the same unit first: 2 m = 200 cm, so the ratio is 200 : 50. Divide both by 50: $4 : 1$. (2:50 fails to convert; 1:4 reverses the order.)

Total parts $= 3 + 5 = 8$, so each part is $\dfrac{240}{8} = 30$ dollars. The larger share is 5 parts: $5\times 30 = 150$ dollars. (Check: 90 + 150 = 240.)

Total parts $= 2 + 3 + 4 = 9$, so each part is $\dfrac{1800}{9} = 200$ dollars. The largest share is 4 parts: $4\times 200 = 800$ dollars.

For every 2 reds there are 5 blues. With 8 reds (which is $4\times 2$), there are $4\times 5 = 20$ blues. (Or $8\times \dfrac{5}{2} = 20$. 13 just adds 8 + 5.)

The angles of a triangle add to $180^\circ$, and the ratio has $2 + 3 + 4 = 9$ parts, so each part is $\dfrac{180}{9} = 20^\circ$. The largest angle is $4$ parts: $4\times 20 = 80^\circ$. ($20^\circ$ is one part only.)

The age difference in ratio terms is $5 - 3 = 2$ parts, which equals the real 6-year gap, so one part is $\dfrac{6}{2} = 3$ years. Sam is 3 parts: $3\times 3 = 9$ years. (15 is the brother's age; 6 is the difference.)

Time $= \dfrac{\text{distance}}{\text{speed}} = \dfrac{5}{15} = \dfrac{1}{3}$ hour. Convert to minutes: $\dfrac{1}{3}\times 60 = 20$ minutes. ($0.33$ minutes left the answer in hours instead of converting.)

Compare the price per gram: $\dfrac{300}{500} = 0.6$ cents per gram versus $\dfrac{440}{800} = 0.55$ cents per gram. The 800 g pack is cheaper per gram, so it is the better value. (Choosing by the lower total price ignores value for money.)

Speed $= \dfrac{180}{3} = 60$ km/h. (Multiplying gives 540; 540 km has the wrong units.)

Distance $= 80\times 2.5 = 200$ km. (Adding gives 82.5; dividing gives 32.)

A horizontal segment means the distance is not changing, so the car is stationary. The graph is horizontal between $t = 2$ and $t = 3$ hours (distance stays at 100 km). (From 0 to 2 hours the line rises, so the car is moving.)

Speed is the gradient: $\dfrac{\text{change in distance}}{\text{change in time}} = \dfrac{100}{2} = 50$ km/h. (100 km/h is the distance; using the full 3 hours gives 33.3; 50 km has the wrong units.)

Time $= \dfrac{45}{3} = 15$ minutes. (Multiplying gives 135; adding gives 48.)

Actual distance $= 8\times 50\,000 = 400\,000$ cm. Convert: $400\,000\text{ cm} = 4000\text{ m} = 4$ km. (40 km is out by a factor of 10.)

Compare as fractions: $3:4 = 0.75$, $7:10 = 0.70$, $2:3 \approx 0.667$, $5:8 = 0.625$, $4:9 \approx 0.444$. The largest is $3:4$. (Bigger-looking numbers do not mean a bigger ratio.)

$12$ cookies need $150$ g, so 1 cookie needs $\dfrac{150}{12} = 12.5$ g. Then $20\times 12.5 = 250$ g. (Or scale by $\dfrac{20}{12}\times 150 = 250$. Inverting the ratio gives 90.)

Convert each side to its real length: $5\times 200 = 1000\text{ cm} = 10$ m and $4\times 200 = 800\text{ cm} = 8$ m. The real area is $10\times 8 = 80$ m². (Scaling the map area by only 200, instead of scaling both lengths, is the common error.)

Rate $= \dfrac{600}{50} = 12$ km/L. ($\dfrac{50}{600}$ gives L/km, the wrong direction; multiplying gives 30,000.)

From the ratio, sand : gravel = 3 : 5. The scale factor is $\dfrac{30}{3} = 10$ (each part is 10 kg). Gravel $= 5\times 10 = 50$ kg. (Inverting gives 18; 10 is just the scale factor.)

$1\text{ km} = 1000\text{ m}$ and $1\text{ h} = 3600\text{ s}$, so $72\text{ km/h} = \dfrac{72\times 1000}{3600} = \dfrac{72\,000}{3600} = 20$ m/s. (Or divide by 3.6: $\dfrac{72}{3.6} = 20$.)

Rate $= \dfrac{240}{8} = 30$ pages per minute, then $30\times 30 = 900$ pages. (30 is just the rate; 7200 multiplies 240 by 30.)

Step 1: students : computers = 5 : 3, so with 60 computers, students $= 60\times \dfrac{5}{3} = 100$. Step 2: teachers : students = 1 : 20, so teachers $= \dfrac{100}{20} = 5$. (100 stops one step early.)