Lesson 7: Proportion and Applications
BasicWhen two ratios are equal, it's a proportion. For example, 2:3 = 4:6. Learn to solve word problems using proportion, e.g. sharing money, mixing juice.
- Two ratios form a proportion = equal ratio values
- Solve proportion: cross multiply a:b = c:d → ad = bc
- Share by ratio: find total parts first, then each part
文體小知識：比例的意義、解比例問題、按比例分配
§Text Passage
1Proportion: an equation with two equal ratios. For example, 2:3 = 4:6. Both sides have the same ratio value (23), so it's true.
2Solving proportion: if there's an unknown x, use cross multiplication. For example, 2:3 = x:6, cross multiply: 2 × 6 = 3 × x, 12 = 3x, x = 4.
3Sharing by ratio: divide an amount by a given ratio. For example, share $100 by 2:3 between A and B. Total parts = 2+3 = 5. Each part = $100 ÷ 5 = $20. A gets 2 parts = $40, B gets 3 parts = $60.
1Summary 重點摘要
When two ratios are equal, it's a proportion. For example, 2:3 = 4:6. Learn to solve word problems using proportion, e.g. sharing money, mixing juice.
2Vocabulary cards Vocabulary
Two equal ratios
Method to solve proportion
Divide by ratio
Sum of all ratio terms
Scale on a map
Equal ratios
Unknown number, usually x
Word problem on ratio
3Step by step Steps
- Block 1: Checking ProportionTo test whether two ratios form a proportion, work out both ratio values and see whether they are equal. Example: 2:3 gives 2 ÷ 3 = 23 and 4:6 gives 4 ÷ 6 = 23; the values match, so 2:3 = 4:6 is a proportion. If the values differ, the two ratios cannot be joined by an equals sign.
- Block 2: Cross MultiplicationCross multiplication is the usual way to solve a proportion: if a:b = c:d, then a × d = b × c. Example: from 3:4 = x:8, cross multiply to get 3 × 8 = 4 × x, which is 24 = 4x; dividing both sides by 4 gives x = 6. This method quickly finds the unknown in a proportion.
- Block 3: Sharing by Ratio (Basic)To share by a ratio, first find the total number of parts, then the value of one part. Example: share $60 in the ratio 1:2, total parts = 1 + 2 = 3, each part = $60 ÷ 3 = $20, so the shares are $20 and $40. Check by adding: $20 + $40 = $60, exactly the total.
- Block 4: Sharing by Ratio (Three Parts)Sharing among three quantities works the same way. Example: share $180 among three people in the ratio 2:3:4, total parts = 2 + 3 + 4 = 9, each part = $180 ÷ 9 = $20, so they get $40, $60 and $80. Check: $40 + $60 + $80 = $180, matching the total, so the sharing is correct.
- Block 5: Application: Mixing DrinksRatios often appear when mixing drinks. Example: juice and water are mixed 1:4; if 50 ml of juice is used, how much water is needed? Let the water be x ml and write 1:4 = 50:x; cross multiply to get x = 4 × 50 = 200, so 200 ml of water is needed. Afterwards check that 50:200 really equals 1:4.
- Block 6: Application: Map ScaleA map scale of 1:100000 means 1 cm on the map stands for 100000 cm in real life, which is 1 km. Example: if the map distance is 5 cm, the real distance = 5 × 1 km = 5 km. Watch the units: 100000 cm = 1000 m = 1 km, so convert to a sensible unit before multiplying to keep the answer correct.
4Diagrams Diagrams
5Worked example Example
Example: Share $120 between Ming and Wah in the ratio 2:3. How much does each get?
- Step 1: find the total number of parts by adding the terms of the ratio: 2 + 3 = 5 parts.
- Step 2: find the value of one part by dividing the total by the number of parts: $120 ÷ 5 = $24.
- Step 3: Ming has 2 parts, so Ming gets $24 × 2 = $48.
- Step 4: Wah has 3 parts, so Wah gets $24 × 3 = $72; check $48 + $72 = $120, which matches the total.
- Answer: Answer: Ming gets $48 and Wah gets $72. To share in a given ratio, first find the total number of parts, then divide the total amount to find one part, and finally multiply by each person's parts; the shares should add back to the total.
6Common mistakes Mistakes
7Quick check Quick check
Q1Solve: 2:5 = x:20, x = ?
Q2Share $90 by 1:2. How much is the bigger share?
Q3Juice:water = 1:5, use 30 ml juice. How much water?
Q4Map scale 1:50000, 4 cm on map = ? real distance
Q5Share $200 by 2:3:5. How much is the smallest share?
8Recap Recap
- Two ratios form a proportion = equal ratio values
- Solve proportion: cross multiply a:b = c:d → ad = bc
- Share by ratio: find total parts first, then each part
- Map scale: map distance × scale denominator = real distance
- Always be clear what corresponds to what