Sec 1 G3 Maths Bukit Timah | Equivalent Ratios and Tables

Did you know? The ratios 3:5, 6:10 and 12:20 all describe the same relationship. The amounts may change, yet the comparison stays exactly the same because both terms were scaled by the same factor. A student who understands that can solve a missing-value table in seconds without guessing or memorising a new equation for every worksheet layout.

For Secondary 1 G3 Mathematics learners in Bukit Timah, equivalent ratios connect Primary School sharing models to Secondary Mathematics’ algebraic way of describing relationships. Students sometimes know how to simplify a ratio but struggle to explain which number to multiply, why the same factor must be used twice, or how to spot a table that is not proportional.

This guide focuses on equivalent ratios, unit relationships and missing entries. It builds on the earlier ratio and sharing quantities guide by examining tables, unknown values, fractional scale factors, algebraic forms and misconceptions that appear when information changes.

Guide contents

Quick answer: multiply or divide both terms by the same nonzero factor
Why 3:5 is not the same as 3:8
A two-column ratio table
Worked example 1: find a missing table entry
Worked example 2: the other column is known
Worked example 3: a missing ratio term
Worked example 4: a fractional scale factor
Why equivalent ratios must use matching units
What is the “one part” approach?
Worked example 5: a total is given instead
Worked example 6: a difference is given
A student should name the known quantity first
Ratios can be written algebraically
Worked example 7: combine two ratio statements

Quick answer: multiply or divide both terms by the same nonzero factor

If A:B = 3:5 and the first quantity A becomes 12, find the second quantity B.

The first term has been multiplied by 4, because 3 × 4 = 12. Multiply the second term by the same 4: 5 × 4 = 20.

Thus 3:5 = 12:20.

A check is to simplify 12:20 back to 3:5 by dividing both terms by 4.

The word “equivalent” means the ratio’s relationship is preserved. Changing only one side or using unrelated factors alters the relationship.

Why 3:5 is not the same as 3:8

Suppose there are three red counters and five blue counters. Red:blue = 3:5. The total number of counters is 3 + 5 = 8.

The ratio of red to all counters is 3:8. That is a different comparison, even though the original red count is still three.

As a fraction of the entire set, red represents 3/8 or 37.5%. As a fraction of the blue count, red is 3/5 or 60% of the blue amount.

Both percentage statements are correct when the denominators are described explicitly. Confusion begins when the student remembers a fraction but forgets what its whole represents.

A two-column ratio table

Suppose red:blue = 3:5. Here are four equivalent pairs:

Red quantity	Blue quantity
3	5
6	10
9	15
12	20

Each row has red divided by blue equal to 3/5, or 0.6.

If you move from the first row to the fourth, multiply both entries by 4. From the second row to the third, multiply both entries by 1.5. You may scale by a non-integer factor when the resulting quantities and problem context permit it.

The student should read the columns’ meanings before calculating. Otherwise, an answer that belongs in the blue column may be placed in the red column.

Worked example 1: find a missing table entry

A table follows the ratio A:B = 3:5. In one row A = 18. What is B?

The scale factor from 3 to 18 is 18 ÷ 3 = 6. Therefore B = 5 × 6 = 30.

Check: 18:30 simplifies to 3:5. Equivalently 18/30 = 0.6, the same as 3/5.

A common mistake is to calculate 18 × 3 ÷ 5 = 10.8, using the ratio in the wrong direction. Ask the learner which original term corresponds to the given 18.

Worked example 2: the other column is known

The ratio A:B remains 3:5. In another row B = 45. Find A.

The second ratio term goes from 5 to 45, so the scale factor is 45 ÷ 5 = 9. The first term is A = 3 × 9 = 27.

Check: 27:45 simplifies by 9 to 3:5.

Both examples use the same principle, but the known column changes. That’s why “multiply by five-thirds” cannot be applied reflexively without identifying which variable is being solved.

One useful checking sentence is, “A is three-fifths of B, so A must be smaller than B when both are positive.” Here 27 < 45.

Worked example 3: a missing ratio term

Complete the equality 3:n = 12:28.

Find the common scale factor from 3 to 12: multiply by 4. Therefore n × 4 = 28, giving n = 7.

Check: 3:7 becomes 12:28 when both values are multiplied by 4.

A fraction route gives the same result: 3/n = 12/28 = 3/7. Therefore n = 7.

Notice that writing an equation is optional for a simple whole-number scale factor. If the structure is visible, direct scaling may be clearer.

Worked example 4: a fractional scale factor

A:B = 4:7. If A = 6 litres, find B.

The scale factor is 6/4 = 1.5. Therefore B = 7 × 1.5 = 10.5 litres.

Check: 6/10.5 = 4/7, so the ratio is preserved.

This is mathematically valid because litres measure a continuous quantity and half-litres make sense. The scale factor need not be a whole number.

But suppose A and B count individual students. If the same information forced B = 10.5 students, the context would be impossible as a count of complete people. The algebraic relation and the real-world constraints must be considered together.

Why equivalent ratios must use matching units

Suppose juice concentrate and water are measured in the ratio 1.5 litres : 600 millilitres. Before simplifying, convert to a common unit.

1.5 L = 1500 mL. The ratio is 1500:600, which simplifies by 300 to 5:2.

An incorrect result of 1.5:600 comes from treating the numbers as though they shared the same unit. They do not.

Once the relationship is written as 5:2, a student can work comfortably with equal parts. If five parts of concentrate represent 1500 mL, each part is 300 mL, and the two water parts represent 600 mL.

For unit basics, see metric length, area and volume conversion.

What is the “one part” approach?

When A:B = 3:5, imagine A consists of three equal parts and B consists of five identical-sized parts.

If A = 36, one part is 36 ÷ 3 = 12. B has five parts, so B = 5 × 12 = 60.

When B = 60, one part is 60 ÷ 5 = 12. A has three parts, so A = 3 × 12 = 36.

The approach works from either known quantity. The scale factor from the simplest ratio is simply the size of one part.

A bar model makes this visible, while an algebraic form makes the same idea succinct.

Worked example 5: a total is given instead

A:B = 3:5, and together A + B = 96. Find both values.

Total parts = 3 + 5 = 8. Each part represents 96 ÷ 8 = 12.

Therefore A = 3 × 12 = 36 and B = 5 × 12 = 60.

Check the total: 36 + 60 = 96. Check the ratio: 36:60 simplifies to 3:5.

This differs from a question that gives one component alone. If the known number were A = 96, dividing by eight would be wrong because 96 would represent only the three A-parts, not the total eight parts.

Worked example 6: a difference is given

Suppose B is 24 more than A and A:B = 3:5.

The difference in parts is 5 − 3 = 2. Two parts represent 24, so one part is 12.

Then A = 3 × 12 = 36 and B = 5 × 12 = 60.

Check: B − A = 24 and 36:60 = 3:5.

These numbers happen to match the preceding total example because the data were chosen consistently: the total is 96 and the difference is 24 for A = 36 and B = 60.

The key lesson is that the given relationship decides whether to add the part counts or subtract them.

A student should name the known quantity first

Three almost identical problems can use the same ratio 2:3 but require different first steps.

Problem A says total = 50. Add the parts: 2 + 3 = 5; one part is 10; the values are 20 and 30.

Problem B says first quantity = 20. Two parts represent 20; one part is 10; the second quantity is 30.

Problem C says difference = 10. The part-count difference is 3 − 2 = 1; one part is 10; the values are again 20 and 30.

The final values agree in this deliberately linked set, but the operations follow distinct interpretations. A tutor can change just one given quantity and see whether the child updates the method.

Ratios can be written algebraically

If A:B = 3:5, then for some common nonnegative scale value k, A = 3k and B = 5k.

If A + B = 96, we can write 3k + 5k = 96. This simplifies to 8k = 96, giving k = 12.

Then A = 36 and B = 60.

If instead B − A = 24, write 5k − 3k = 24. This gives 2k = 24 and again k = 12.

The algebra isn’t a different subject from the bar model. Both describe how many equal-sized parts each quantity contains.

Worked example 7: combine two ratio statements

Suppose A:B = 2:3 and B:C = 4:5. Find A:B:C.

We cannot write 2:3:5 immediately because the middle B has three parts in the first statement and four in the second. They need a common scale.

Make the B term equal to twelve parts. Multiply 2:3 by 4 to obtain A:B = 8:12. Multiply 4:5 by 3 to obtain B:C = 12:15.

Combine: A:B:C = 8:12:15.

Check A:B = 8:12 = 2:3 and B:C = 12:15 = 4:5. Both conditions are satisfied.

This is a useful extension once simple two-column ratios are secure. The shared quantity is the anchor.

The algebra of equal ratios

Writing 3:5 = 12:20 is equivalent to saying 3/5 = 12/20, because both fractions describe the first quantity compared with the second.

You can verify by products: 3 × 20 = 60 and 5 × 12 = 60. For nonzero denominators, equality of the two fractions follows.

But cross-products are not a substitute for understanding which terms refer to which quantities. A student who swaps columns can produce a neat multiplication sentence that represents the wrong relationship.

Encourage the learner to label each fraction “first compared with second” before choosing a route.

When a ratio table is not proportional

Imagine a fictional service with a $4 fixed fee plus $3 per unit purchased.

For one unit, the total cost is $7. For two units, it is $10. For three units, it is $13.

The cost-to-unit ratios are 7:1, 10:2 and 13:3, which are not equivalent. Doubling the units from one to two did not double the total cost from $7 to $14.

Why? The fixed $4 charge is paid only once. The variable cost scales, but the total includes an unchanged component.

A student should therefore not use proportional ratio scaling on every table that contains a consistent pattern. Direct proportionality requires the relevant quotient to remain constant.

How to test whether a table preserves one ratio

Suppose a table has pairs (2, 6), (4, 12) and (6, 18). For each pair, second ÷ first = 3. That constant quotient is consistent with the relationship second = 3 × first.

Now consider (2, 6), (4, 12), (6, 20). The first two give quotient 3, but 20/6 ≈ 3.333. The third row breaks the constant ratio.

The error may be an incorrectly recorded value or evidence that the proposed ratio model isn’t valid. Do not force all rows to agree by changing the data silently.

A good Mathematics answer identifies which row conflicts with the proposed relationship.

What happens when quantities are increased by the same amount?

Suppose A:B = 3:5 and the actual amounts are 36 and 60.

Add 12 to each: new amounts become 48 and 72. The ratio is now 48:72 = 2:3, not 3:5.

The difference stays 24, since both quantities increased by twelve, but the ratio changes.

Equivalent ratios are preserved by multiplying both entries by the same nonzero scale factor, not generally by adding the same number to both.

This is a powerful counterexample for students who confuse equal differences with equal relationships.

Ratios and percent of a whole

For positive A:B = 3:5, total parts are eight. A represents 3/8 of the total, or 37.5%. B represents 5/8, or 62.5%.

But A compared with B is 3/5, meaning A is 60% of B.

The percentage differs because the denominator is different. Both statements are correct when the comparison is properly labelled.

If the child says A is “three-fifths of the total,” ask them to draw three A-parts plus five B-parts. The diagram makes the total eight parts unmistakable.

A complete table can be checked two ways

Consider an extended ratio table with A:B = 4:7.

A	B	Check
4	7	original
8	14	double both
12	21	triple both
20	35	multiply both by five
6	10.5	multiply both by one and a half

First check by simplifying each ratio back to 4:7. Second check by dividing B by A: each quotient should equal 7/4 = 1.75.

The latter row shows a decimal amount. It is mathematically valid for continuous measurements, though it would not describe discrete objects if a half-object were forbidden.

The context must support the numerical values as well as the relationship.

Four distinct mistake patterns

Reversed scaling: Given A:B = 3:5 and A = 18, the student multiplies 18 by 3/5 to get 10.8 rather than 30. They have applied the ratio in the wrong direction.

Part-to-whole confusion: A:B = 3:5 is treated as A being three-fifths of the total. The denominator should be eight total parts.

Unmatched units: 1.5 L:600 mL is simplified without converting to a common unit.

False proportionality: A fixed-fee table is assumed to follow a constant total-price-per-unit ratio, although the extra charge prevents it.

Each error needs a targeted correction. A second, changed example should test whether the reason has been understood.

Fifteen-question mini diagnostic

A. Complete 3:5 = 12:__.
B. Complete 3:5 = __:45.
C. Complete 3:n = 12:28.
D. If A:B = 3:5 and A = 18, find B.
E. If A:B = 3:5 and B = 45, find A.
F. Simplify 24:40.
G. Convert 1.5 L:600 mL into simplest form.
H. If A:B = 3:5 and total is 96, find A.
I. In Question H, find B.
J. If A:B = 3:5 and B exceeds A by 24, find A.
K. If red:blue = 2:3, what fraction of the total is red?
L. If A:B = 4:7 and A = 6 litres, find B.
M. If A:B = 2:3 and B:C = 4:5, find A:B:C.
N. Does adding ten to both parts generally preserve a ratio?
O. A fixed $4 fee plus $3 per item gives costs $7 for one item and $10 for two. Are the total-cost-to-quantity ratios equivalent?

Answers with their reasoning

A. 20. Both parts multiply by four.

B. 27. The second part scales by nine.

C. 7. Both original terms multiply by four to give 12:28.

D. 30. One part is six.

E. 27. One part is nine.

F. 3:5. Divide both terms by eight.

G. 5:2. Convert 1.5 L to 1500 mL first.

H. 36. Eight total parts represent 96, so one part is twelve.

I. 60. Five parts of twelve.

J. 36. The difference of two parts represents 24.

K. 2/5. Red and blue together account for five parts.

L. 10.5 litres. Multiply seven by 1.5.

M. 8:12:15, using a common middle term of twelve.

N. No. Equal addition generally changes the ratio.

O. No. 7/1 is not equal to 10/2; the fixed charge prevents constant total cost per unit.

A useful marking technique is to ask students to explain which number represents one part before giving them the answer key.

A ten-minute parent diagnostic

Show three rows of the same ratio with one unknown entry. Let the child explain the matching scale factor and fill in the blank.

Next present a similar-looking table containing a fixed cost. Ask whether doubling the quantity doubles the total; the student should notice the added fee.

Then give a total-sharing problem and a one-component-known problem. The ratio is unchanged, but the relevant part count differs.

Finish with a fractional scale factor. If quantities are measured continuously, a result such as 10.5 litres is entirely reasonable. Ask when a decimal result would instead violate the story’s discrete-count conditions.

How a 90-minute small-group lesson can repair ratio thinking

A full lesson need not spend ninety minutes on repeated equivalent-ratio sums. The eduKateSG reference tutorial describes a structure including retrieval, concept explanation, guided work, independent application, mixed practice and error review.

An appropriate ratio session can begin with four visual bar examples and a short number-sense check. Then move to a three-column table and let the tutor ask which quantity is the common anchor.

After guided practice, remove the model. The student should solve a changed table and explain why an unrelated fixed-fee table is not proportional. Finally, revisit the key mistake after a delay.

The important outcome is not speed on the example itself; it is independent identification of a preserved relationship.

A progress checklist for Bukit Timah families

Early progress: the child can scale both ratio terms by the same number and check the equivalence.

Intermediate progress: the child can move in either direction, solve when a total or difference is given, and handle a fractional scale factor.

Stronger progress: the child can connect ratio statements with an equation, combine two statements sharing a middle quantity, and recognise a table whose relationship is not proportional.

These are observable milestones. They provide a clearer discussion with a tutor than a vague report that “ratios are better now.”

Choosing Secondary 1 G3 Mathematics support in Bukit Timah

Ask how the tutor diagnoses whether a student is mixing up part-to-part and part-to-whole comparisons, reversing the scale factor or forgetting to convert units. The correct intervention depends on the source of error.

The immutable eduKateSG Secondary 1 Mathematics small-group reference describes three-student tutorials near Sixth Avenue MRT with close feedback and a 90-minute learning cycle. The Bukit Timah G3 Mathematics tuition overview gives local programme context. Confirm schedules, places and your child’s school topic sequence directly.

The official MOE G2 and G3 Mathematics syllabus provides the curriculum framework for rational-number ratios, equivalent forms and practical ratio questions.

A decimal ratio table that still preserves exact proportions

Imagine a fictional mixture with quantities A:B = 7:9, both measured in kilograms. The first amount is 12.25 kg. What should the second amount be? One ratio part is 12.25 ÷ 7 = 1.75 kg. Nine parts correspond to 9 × 1.75 = 15.75 kg. Together they total 28 kg.

Check the comparison without relying on the total: 12.25/15.75 = 7/9. You can see the equivalence by scaling both quantities down by 1.75. The decimals do not make this a new kind of relationship; each part has simply become a non-integer amount.

Now imagine the same ratio describes a count of seven red tokens for every nine blue tokens. Scaling by 1.75 would produce 12.25 and 15.75 individual tokens, which do not make sense as counts of complete objects. The proportional mathematics is correct for continuous mass, while the discrete-token story would need a permissible whole-number scale factor. Ask which values the context allows.

A table can grow by equal differences without preserving a ratio

Consider paired values (5, 8), (10, 16) and (15, 24). Each time the first number increases by five, the second increases by eight. The quotients 8/5, 16/10 and 24/15 all equal 1.6. This is a directly proportional relationship and an equivalent-ratio table.

Now consider (5, 8), (10, 13) and (15, 18). Both columns still increase by five between rows. But their quotients are 8/5 = 1.6, 13/10 = 1.3 and 18/15 = 1.2. The numbers form an orderly pattern, yet the ratios are not equivalent.

The second table follows a rule of adding three to the first value, not of multiplying by one constant ratio factor. It is a useful bridge to function tables: a consistent addition pattern and a consistent multiplicative relationship are not the same concept.

When a child says a table ‘looks regular,’ ask whether *differences* or *quotients* remain constant. That single question can prevent a wrong proportionality assumption.

An optional proof check using two fractions

Suppose a learner says that 14:21 and 22:33 are equivalent. Both simplify to 2:3, so the statement is true. Another verification is 14/21 = 2/3 and 22/33 = 2/3.

For nonzero second terms, we can also check cross-products: 14 × 33 = 462 and 21 × 22 = 462. Equal products confirm the fraction equality. But this technique is only meaningful after each number has been correctly associated with the same first or second quantity.

Replace 22:33 with 22:30. The cross-products are now 14 × 30 = 420 and 21 × 22 = 462, so the relationships differ. The method exposes a non-equivalent pair even when both terms have increased numerically.

Students should be encouraged to explain what is held constant in the relationship. Multiplication checks are useful tools, but they are not substitutes for a labelled, correctly interpreted ratio.

Frequently asked questions

Is an equivalent ratio the same as an equal difference?

No. Equivalent ratios keep the same multiplicative relationship. Equal differences can describe very different ratios.

Does the scale factor have to be a whole number?

No. For continuous quantities, a factor such as 1.5 is valid. Real-world discrete counts may restrict the possible scaled values.

When do I add the ratio terms?

When the given amount represents the combined total of all parts. If one component alone is known, use its own part count.

Why is a table with a fixed charge not usually a ratio table?

Because the fixed component does not scale when the number of items changes. The overall quotient of cost divided by item count therefore varies.

How do I check a missing value?

Simplify the completed pair back to the original ratio or compare their two quotients using the same column order.

What if my child can simplify but cannot solve words?

Ask them to label each quantity, identify whether the given amount is a total, difference or component, and explain the size of one part before calculating.

The next time a ratio table has a blank, ask the child one question: “What happened to the known number, and must that same thing happen to its partner?” The relationship—not the blank—is the real mathematics.

Secondary 1 G3 Mathematics Tutor Bukit Timah | Equivalent Ratios, Tables and Missing Values