Example 2: Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add?
Answer
30 spoons of sugar for the 18 extra glasses.
glasses : spoons must stay proportional 6 : 10 :: 18 : ? First term: 6 → 18, factor = 18 ÷ 6 = 3 Second term must change by 3 as well: 10 × 3 = 30 6 : 10 :: 18 : 30
Why the factor must be the same: Sweetness is the amount of sugar per glass. In the first vessel that is 10⁄6 spoons per glass. If the number of glasses is tripled but the sugar is not, each glass gets a third as much sugar and the drink tastes weak. Tripling both keeps 30⁄18 = 10⁄6 — the same sweetness.
Tip: Altogether Kesang will then have made 6 + 18 = 24 glasses using 10 + 30 = 40 spoons, and 24 : 40 also simplifies to 3 : 5. The whole batch is as sweet as the first vessel.
Q4.
How can we find the factor of change in the ratio?
Answer
Divide the new term by the old term of the same quantity.
factor = new first term ÷ old first term = 18 ÷ 6 = 3
The factor need not be a whole number. If the first term had gone from 6 to 9, the factor would be 9 ÷ 6 = 3⁄2, and the second term would have to be multiplied by 3⁄2 as well.
Q5.
Example 3: Nitin and Hari were constructing a compound wall around their house. Nitin was building the longer side, 60 ft in length, and Hari was building the shorter side, 40 ft in length. Nitin used 3 bags of cement but Hari used only 2 bags of cement. Nitin was worried that the wall Hari built would not be as strong as the wall he built because she used less cement. Is Nitin correct in his thinking?
Answer
No, Nitin is not correct. The two walls are equally strong.
Why it happens: Strength depends on how much cement goes into each foot of wall, not on the total number of bags. Both are building at 1 bag for every 20 feet. Hari used fewer bags only because she had less wall to build. Comparing the totals alone (3 against 2) compares two things of different sizes; the ratio compares them fairly.
Q6.
Example 4: In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below. ______ : ______ Is the teacher-to-student ratio in your school proportional to the one in my school?
Answer
This one needs your own school's figures, but here is exactly how to settle it.
Given school: 5 : 170, HCF = 5 → 1 : 34 (one teacher for every 34 students)
Count the teachers and the students in your school, write the ratio, and reduce it. Then compare:
If your simplest form is also 1 : 34, the two ratios are proportional.
If it is 1 : 25, each teacher in your school handles fewer students.
If it is 1 : 40, each teacher handles more.
Check it yourself: Suppose your school has 12 teachers and 408 students. Then 12 : 408 has HCF 12 and reduces to 1 : 34 — proportional to 5 : 170, even though both numbers are much larger. Cross multiplication says the same thing: 5 × 408 = 2040 = 170 × 12.
Q7.
Example 5: Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard? ______ : ______ Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard?
Answer
Measure with a metre scale and reduce the ratio you get. A very common classroom size is 240 cm by 120 cm.
Width : height = 240 : 120, HCF = 120 → 2 : 1
To draw a proportional rectangle in your notebook, pick any convenient factor and multiply both terms by it:
Factor
Width
Height
Ratio
× 4 cm
8 cm
4 cm
2 : 1
× 5 cm
10 cm
5 cm
2 : 1
× 6.5 cm
13 cm
6.5 cm
2 : 1
Tip: If your blackboard measures, say, 210 cm by 90 cm, the ratio is 210 : 90 = 7 : 3, and a 14 cm by 6 cm rectangle is proportional to it.
Q8.
Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
Answer
They will be of different sizes but the same shape — every one of them is a scaled copy of the blackboard.
Why it happens: Each of you was free to choose the factor, so the sizes differ. But nobody was free to change the ratio, so the shape does not differ. Place any two of the drawings one on top of the other with a corner matched, and the diagonals will lie along the same line — the sure sign of two rectangles with equal ratios.
Check it yourself: A rectangle that does not match has a diagonal at a visibly different slant. That is a quick way to spot the odd one out in the whole class without measuring anything.