Problem
Water height from base area
The same 42 cubic centimetres of water goes into two containers. Container A is a box with a 3 cm by 2 cm base. Container B is a prism with a triangular base. Find the ratio of the two water heights.
Reasoning · Grade 6-1 Volume of Solid Figures
Your answer
Watch out A wider base seems to hold more depth, but with equal water it is shallower.
How to solve
Strategy Identify Subproblems — The water in a prism is itself a prism, so its volume is base area times water height. That single sentence turns the question into three easy subproblems: find A's base area, find B's base area, and in each case ask what height multiplied by that base area gives 42. Drawing the level surface across each container is what makes it clear that the water is a short prism sitting inside a tall one. Doing the two containers separately and then comparing also reveals the pattern worth keeping: the same water in a wider base stands lower, so the heights come out in the reversed ratio of the base areas.
1STEP 1
See that the water is itself a prism
The water is a prism too: base times height.
(volume of water) = (base area) × (water height)
A prism is the same shape stacked up over and over. Pouring water in just decides how many of those identical layers you get, so the volume rule for the whole container works for the water part too.
7.G.B.6Draw A DiagramThe water itself is a prism, so its height is the poured volume shared out over the base area.
Why?
A prism's volume is its base area multiplied by its height, so knowing the volume and the base hands over the height.
🧱A prism is its base stacked upA prism is one flat face repeated all the way up, so its volume is the base area multiplied by the height.
Why?
With the volume fixed, a base twice as large gives a height half as tall, so the heights run opposite to the base areas.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
2STEP 2
Find the base area of container A
Container A's base is 6 cm².
3 × 2 = 6 cm²
Area of a rectangle is length times width — the number of 1 cm squares that fit on the floor of the box.
6.G.A.1Identify Subproblems3STEP 3
Find the water height in container A
So the water stands 7 cm deep.
6 × (height) = 42 → (height) = 42 ÷ 6 = 7 cm
Each 1 cm layer of water in A holds 6 cubic centimetres, so 42 cubic centimetres is 7 such layers. Dividing the volume by the base area counts the layers.
7.G.B.6Identify Subproblems4STEP 4
Find the base area of container B
Container B's base is 14 cm².
7 × 4 ÷ 2 = 28 ÷ 2 = 14 cm²
The right-angle mark is what makes this easy: it certifies that the 4 cm really is the perpendicular height on the 7 cm edge, which is the only height the triangle formula accepts. Two copies of the triangle fit together into a parallelogram of base 7 and height 4, which has the same area as a 7 by 4 rectangle - so the triangle is half of 28, wherever along the edge the far vertex sits.
6.G.A.1Identify Subproblems5STEP 5
Find the water height in container B
So the water stands 3 cm deep.
14 × (height) = 42 → (height) = 42 ÷ 14 = 3 cm
A 1 cm layer in B holds 14 cubic centimetres — more than twice what a layer in A holds — so the same water makes far fewer layers and stands much lower.
7.G.B.6Identify Subproblems6STEP 6
Write the ratio of the two heights
The ratio of heights is 7 : 3.
(height in A) : (height in B) = 7 : 3
A ratio compares two measurements of the same kind. Both numbers here are heights in centimetres, so the units cancel and the comparison 7 to 3 is a pure number.
6.RP.A.1Analyze The Units7STEP 7
Notice the pattern: the heights reverse the base areas
The heights are the base areas reversed.
(base area A) : (base area B) = 6 : 14 = 3 : 7 ⟹ (height A) : (height B) = 7 : 3
When two things multiply to a fixed total, making one bigger forces the other smaller by the same factor. Spotting that lets you write the ratio straight from the base areas without ever computing either height.
7.RP.A.2Look For A PatternAnswer
7 : 3
42 ÷ 6 = 7, 42 ÷ 14 = 3
The units are right at every stage: square centimetres times centimetres gives cubic centimetres, and 6 x 7 = 42 and 14 x 3 = 42 both return the poured volume exactly. The heights are believable — 7 cm and 3 cm are both under the 10 cm the containers stand, so neither overflows, which is what the problem intends. The direction is right too: B has the larger floor (14 square centimetres against 6), so the same water should spread out and stand lower there, and 3 cm is indeed less than 7 cm. Finally, the ratio 7 to 3 is exactly the reverse of the simplified base-area ratio 3 to 7, which is the check that the inverse relationship was used the right way round rather than upside down.
Takeaway
Pour the same water onto a wider floor and it stands lower — the heights come out in exactly the reversed ratio of the base areas.
- See that the water is itself a prism
- Find the base area of container A
- Find the water height in container A
- Find the base area of container B
- Find the water height in container B
- Write the ratio of the two heights
- Notice the pattern: the heights reverse the base areas