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Exchange-rate quotes need the same direction before you compare their numbers. A quote of 1 A = 1.25 B describes how many units of currency B correspond to one unit of currency A. The equivalent reverse quote is 1 B = 0.8 A. The numbers 1.25 and 0.8 describe the same relationship in opposite directions, so the larger number is not automatically a better offer.
For a G1 Mathematics exchange question, write the currency you start with and the currency you need. Translate each quote into the amount received for the same starting amount, or the amount paid for the same target amount. Then apply any stated charges in the stated order. Compare final quantities with matching currencies and matching conditions.
Contents
Sections 1–5
Sections 6–10
- 5. Two differently written quotes can be equivalent
- 6. A larger rate can be better or worse depending on its meaning
- 7. Keep the starting amount the same when comparing receipts
- 8. Keep the target amount the same when comparing costs
- 9. A fixed charge in the starting currency changes the convertible amount
Sections 11–15
Sections 16–20
Sections 21–22
Suppose you exchange 200 A into B at the fictional rate 1 A = 1.25 B, with no charge. You receive 200 × 1.25 = 250 B. Using the equivalent quote 1 B = 0.8 A, you calculate 200/0.8 = 250 B. Multiplication and division both appear, but each follows the direction of the quote.
Now suppose another fictional quote gives 1 A = 1.2 B. For the same 200 A, it gives 240 B before charges. The first quote gives more B for that particular conversion. If the task instead starts with B and asks how much A it can obtain, the comparison must be read in that direction.
The useful question is not “Is this rate number larger?” It is “How much of the requested currency results from the same input under these conditions?” That question attaches the numerical comparison to the actual quantity being found.
Every rate and charge in this guide is invented for mathematics practice. A and B are fictional currency labels. The examples are not current market quotes or recommendations about real transactions.
The SEAB G1 Mathematics syllabus for 2027 includes money exchange among its real-world contexts. Such questions combine rates, proportional reasoning, units and interpretation. The operation becomes clearer when the two currency roles remain visible throughout the working.
This guide focuses on quote direction. It does not attempt to explain financial markets or predict currencies. The classroom skill is to read what one numerical statement means, convert it into a useful relationship, and compare like quantities.
For the broader subject route, use what G1, G2 and G3 Mathematics mean. Select practice according to the methods already taught. A difficult-looking exchange table often becomes manageable once the starting currency and target currency are named.
The guide to reading quantities before calculating provides the foundation for this approach. A rate is not just a decimal. It connects two differently labelled amounts, and reversing the labels changes how the decimal is used.
Begin with one quote and no charges. Once that is secure, compare two quotes, then add a fixed charge or a target amount. Each new condition should be attached to an understood relationship rather than hidden inside a memorised instruction to multiply or divide.
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The statement 1 A = 1.25 B says that one unit of A corresponds to 1.25 units of B in the model. You may scale both amounts together: 2 A correspond to 2.5 B, 10 A to 12.5 B, and 200 A to 250 B.
The equality does not say that 1 B equals 1.25 A. That would be a different relationship. To reverse the quote, divide both sides by 1.25: 0.8 A = 1 B. Reordering the sides gives 1 B = 0.8 A.
A common error preserves the number while swapping the currency names. That changes the relationship instead of expressing it in reverse. A correct reciprocal quote changes both the direction and the numerical rate.
Write the complete equality before extracting a rate. “1.25” alone leaves its direction unspecified. “1.25 B per A” records the target amount for each unit of the starting currency. This is the information needed for a dimensional check.
The ratio-table guide develops the same idea of scaling linked quantities together. An exchange quote is a particularly clear example because the units on the two sides are visibly different.
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Suppose the rate is 1.25 B per A and you start with 200 A. Multiplying gives 200 A × 1.25 B per A = 250 B. The A unit in the input matches the per-A denominator of the rate, leaving B as the output unit.
You can explain this without formal unit cancellation: every A gives 1.25 B, so 200 such units give 200 lots of 1.25 B. The arithmetic follows directly from repeated proportional scaling.
Check the size of the result. Because each A corresponds to more than one B, the numerical amount in B is greater than the numerical amount in A. This is a check within the stated quote, not a claim that one currency amount is inherently more valuable because its written number is larger.
If the rate were 0.75 B per A, 200 A would give 150 B. A smaller numerical output can still be the correct equivalent amount because the units differ. Do not compare 200 A and 150 B as though they were counts of the same object.
The final answer should include the received currency. “250” is incomplete when a question contains several currencies or several stages. “250 B before charges” names both the unit and the stage of the calculation.
One practical paper check is to place a small arrow from the starting currency to the target currency beside the calculation. Under it, write what one starting unit produces. If the supplied quote points the other way, either invert it explicitly or use the corresponding division. This short annotation keeps the decision visible without adding a long explanation to every line. Remove the annotation once the learner can interpret either quote direction reliably.
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Suppose the quote is given as 1 B = 0.8 A, while you still start with 200 A and need B. Each unit of B costs 0.8 A in this model. The number of B units obtainable is therefore 200/0.8 = 250.
The division asks how many groups of 0.8 A fit into 200 A. Each such group corresponds to one B. This interpretation is more useful than memorising that one named currency always requires division.
The same quote can also support multiplication in the other direction. Starting with 250 B, multiply by 0.8 A per B to obtain 200 A. The operation depends on which quantity is given and which is requested.
A helpful check is to convert back using the same ideal no-charge relationship. If 200 A becomes 250 B and 250 B becomes 200 A, the directions are consistent. If the return gives a different amount, inspect whether you reversed the rate correctly or used the wrong operation.
This reversibility belongs to the simplified model of one reciprocal relationship without charges or differing transaction quotes. A later section explains why a problem with separate outward and return rates must be handled using the rates actually supplied, rather than forcing the ideal return to match.
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Consider Quote P: 1 A = 1.25 B. Quote Q: 1 B = 0.8 A. To compare them for conversion from A to B, express both as B received per A. P already gives 1.25. For Q, calculate 1/0.8 = 1.25 B per A.
The quotes are equivalent before any charges. Comparing the bare numbers 1.25 and 0.8 would miss that fact. The difference in numerical size comes from the direction of the description, not from a difference in the underlying conversion.
You can also use a common starting amount instead of a unit rate. With 100 A, P gives 125 B and Q gives 100/0.8 = 125 B. This method is often easier for a learner who finds reciprocal notation abstract.
Both approaches are valid. The unit-rate method normalises each quote to one starting unit. The common-input method normalises each calculation to the same chosen starting amount. Each creates a like-for-like comparison.
If a table gives 100 A = 125 B rather than 1 A = 1.25 B, divide both amounts by 100 first or scale directly from the stated pair. The size of the quoted batch does not alter the relationship, provided the problem describes a proportional rate without additional conditions.
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Suppose two quotes both state B received per A: P gives 1.25 B per A and R gives 1.2 B per A. For the same amount of A, P produces more B before charges. Here the larger rate is favourable for the stated A-to-B task.
Now express prices as A required per B. P requires 0.8 A per B. A competing quote requires 0.83 A per B. For buying the same amount of B, 0.8 requires less A. Here the smaller rate is favourable for that target-purchase task.
There is no contradiction. One rate measures output per input; the other measures input per output. Reversing the measure reverses the direction of a simple numerical comparison.
Avoid writing “the higher exchange rate is always better.” The word higher is incomplete until the numerator, denominator and transaction direction are identified. A clear answer names the output or cost being compared.
For example: “At the same starting amount of A and with no charges, P gives more B.” Or: “For the same target amount of B, P requires less A.” These sentences state the mathematical advantage without extending it into an unsupported judgement about every possible transaction or real provider.
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An invented question offers P at 1.25 B per A and R at 1.2 B per A. A learner converts 200 A using P and 300 A using R, obtaining 250 B and 360 B, then says R is better because 360 exceeds 250.
That comparison changed the input as well as the quote. The larger receipt partly reflects exchanging more A. To compare the rates for a fixed-input task, use the same starting amount.
For 200 A, P gives 250 B and R gives 240 B. For 300 A, P gives 375 B and R gives 360 B. P gives more B at either common input under the simple no-charge model.
This is an example of controlling the comparison. State what is held fixed and what changes. In this case, the starting amount and the charge assumptions stay fixed, while the conversion quote changes.
The guide to comparing pack offers and conditions develops the wider habit of putting offers on a comparable basis. The quantities differ here, but the reading discipline is the same: a larger final number does not settle a comparison if the inputs or conditions also changed.
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Some questions specify the required amount of B rather than the available amount of A. Suppose the target is 500 B. At 1.25 B per A, the required A is 500/1.25 = 400 A. At 1.2 B per A, it is 500/1.2, approximately 416.67 A before any stated rounding or charges.
The comparison now concerns A paid for the same B target. Do not compare receipts from different starting amounts when the question has fixed the desired output.
If a quote is already written as A per B, multiply the target B by that rate. At 0.8 A per B, 500 B require 400 A. This is the same calculation as dividing by the reciprocal rate of 1.25 B per A.
Record whether an answer is exact in the model or rounded for presentation. The value 500/1.2 does not terminate at two decimal places. Keep the expression intact during further work and apply the question’s requested final convention.
The guide to early rounding in reverse percentages explains the related precision issue. Recovering a required input by division can be correct in structure and still become inaccurate if a necessary intermediate value is shortened too soon.
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SECTION 10 OF 22
9. A fixed charge in the starting currency changes the convertible amount
Suppose a fictional service charges 4 A from a total budget of 200 A before converting the balance at 1.25 B per A. Only 196 A are converted. The receipt is 196 × 1.25 = 245 B.
Converting all 200 A first and then subtracting 4 B would give 246 B. That is a different charge model because both the currency and the stage of the fee changed. The unit attached to the charge matters.
A simple record is: total budget, starting-currency charge, amount converted, conversion result. It makes the subtraction’s position visible. You do not need to memorise a combined formula before understanding those stages.
Check the result by valuing the deducted 4 A at the same rate. It corresponds to 5 B, so the no-charge receipt of 250 B is reduced to 245 B. This confirms the effect of the charge within the stated model.
Do not assume that every real or classroom fee is deducted this way. The example explicitly defines the order. Another question may add a charge on top of the available budget or deduct it from the received currency. Read the statement rather than importing a familiar transaction convention.
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Now suppose the service converts all 200 A at 1.25 B per A and then deducts a charge of 4 B. The gross receipt is 250 B and the net receipt is 246 B. This differs from the previous starting-currency charge of 4 A.
The matching numerical fee of 4 does not make the fees equivalent. One is measured in A and the other in B. Under the stated rate, 4 A corresponds to 5 B, so it produces a larger reduction in the final B receipt than a fee of 4 B.
If the question compares two offers with different fee currencies, calculate the final requested amount under each complete set of conditions. Comparing only the rate or only the fee number leaves part of the problem out.
For example, one offer may have a higher gross rate but a larger fixed charge. Whether it gives more B can depend on the starting amount. This is why the amount being exchanged belongs in the comparison.
The fixed-starting-amount and proportion guide explains why a fixed addition or subtraction changes a simple proportional model. A net receipt after a fixed fee is not necessarily obtained by scaling one small transaction’s net receipt in direct proportion.
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Suppose a model converts 200 A at 1.25 B per A, then charges 2% of the gross B receipt. The gross amount is 250 B, the charge is 5 B, and the net receipt is 245 B.
If another model deducts 2% of the starting 200 A before conversion, the fee is 4 A, leaving 196 A. At the same rate, the result is also 245 B. These two particular proportional procedures agree because the same percentage multiplier and conversion multiplier commute in the ideal model.
That agreement does not mean every fee order can be ignored. Add a fixed charge or change the percentage base and the result may change. For example, deducting a fixed 4 A and then charging a percentage of the converted balance is a different process from charging that percentage on the full original conversion.
Write “2% of ___” before calculating. The blank should be filled with the quantity named in the problem, not automatically the largest number on the page. A percentage is a relationship to a base.
This section is a mathematical comparison of explicitly defined procedures. It does not describe an actual provider’s fee policy. When a school question supplies a policy, treat that text as part of the data and keep the stated order visible in the working.
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Consider two fictional offers for converting A into B. Offer P gives 1.25 B per A and deducts 10 B after conversion. Offer R gives 1.2 B per A with no charge. For a starting amount of 100 A, P gives 125 − 10 = 115 B, while R gives 120 B.
For 300 A, P gives 375 − 10 = 365 B, while R gives 360 B. The preferred receipt changes because the fixed fee stays the same while the rate advantage grows with the amount exchanged.
At 200 A, both give 240 B: P gives 250 − 10, and R gives 200 × 1.2. This is the equal-receipt point in the stated model. A question may ask you to find it by inspection, a table or algebra, depending on the methods taught.
The comparison is therefore amount-specific. “P has the higher rate” is true but incomplete. “P gives more net B for 300 A under these charges” answers the actual question.
This example also shows why a unit-rate comparison alone can be insufficient after fixed charges. The gross conversion remains proportional, but the net receipt includes a fixed subtraction. Calculate the full net amount for the stated input rather than treating the advertised numerical rate as the whole offer.
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In the ideal reciprocal model, 200 A become 250 B and then return to 200 A using 0.8 A per B. But a problem may supply a different rate for the return transaction, or include charges at one or both stages.
Suppose the outward rate is 1.25 B per A and the stated return rate is 0.78 A per B. Converting 200 A gives 250 B; converting that entire amount back gives 250 × 0.78 = 195 A. The model’s round trip does not return the original amount.
Do not replace the supplied return quote with the reciprocal 0.8 just to force the amounts to match. The question has described two different conversion relationships. Each must be used in its stated direction.
If the question instead explicitly gives one reciprocal no-charge rate for both directions, use that simpler model. The correct procedure depends on the supplied conditions, not on a universal assumption that return conversions must always match or must always differ.
The final sentence should identify the stage: “195 A after the return conversion.” That prevents the intermediate 250 B from being mistaken for the final answer and keeps currencies from being mixed in a subtraction. A difference such as 200 A − 195 A is meaningful; subtracting 250 B directly from 200 A is not.
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A three-currency exercise gives 1 A = 1.25 B and 1 B = 2 C. Starting with 80 A, the first conversion gives 100 B and the second gives 200 C. The combined rate is 1.25 × 2 = 2.5 C per A.
If the second quote is instead written 1 C = 0.5 B, you can calculate 100/0.5 = 200 C. The quote direction changes the operation at that stage but not the final result for an equivalent relationship.
Before combining rates, make sure the intermediate currency matches. A rate in B per A can connect to a rate in C per B. Two rates both written per A do not automatically form a useful sequential conversion by multiplication.
Retain the intermediate currency label even if you combine the calculation into one expression. It helps explain why a factor is multiplied or inverted. Without labels, a chain of decimals can conceal a reversed stage.
Apply charges at the stage where the problem places them. A fixed charge in B between conversions must be deducted from the B amount before converting the balance to C. Deducting the same numerical amount in C at the end would describe a different fee. The linked-unit record keeps these distinctions visible.
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Question A: A quote states 1 A = 1.4 B. Find the B received from 150 A with no charge. Write the equivalent quote for 1 B in terms of A.
Question B: P states 1 A = 1.25 B. Q states 1 B = 0.8 A. Compare the B received from 120 A. Explain why comparing 1.25 directly with 0.8 is not a valid receipt comparison.
Question C: A service charges 3 A from a total 100 A budget, then converts the balance at 1.2 B per A. Find the receipt. Compare it with converting all 100 A and then deducting 3 B.
Question D: P gives 1.3 B per A and deducts 12 B afterwards. R gives 1.25 B per A with no charge. Compare net receipts for 100 A and 300 A.
Question E: The target is 240 B. At a rate of 1.2 B per A with no charge, find the A required. Explain why multiplying 240 by 1.2 would use the quote in the wrong direction.
Question F: Starting with 200 A, convert at 1.25 B per A and then return at 0.76 A per B. Find the final A. Use the supplied return quote rather than assuming it is reciprocal.
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For A, 150 × 1.4 = 210 B. The reverse quote is 1 B = 1/1.4 A = 5/7 A. Retain the fraction if another calculation follows; it expresses the reciprocal exactly in this model.
For B, P gives 120 × 1.25 = 150 B. Q gives 120/0.8 = 150 B. The quotes are equivalent. The decimals have different directional units and cannot be ranked as though they measured the same rate.
For C, the first model converts 97 A, giving 116.4 B. The second converts 100 A to 120 B and then deducts 3 B, giving 117 B. The fees share a numerical value but not a currency or stage.
For D, at 100 A, P gives 118 B and R gives 125 B. At 300 A, P gives 378 B and R gives 375 B. The fixed charge makes the comparison depend on the starting amount.
For E, 240/1.2 = 200 A. Each A produces 1.2 B, so 200 such units produce the target 240 B. For F, the outward conversion gives 250 B and the return gives 250 × 0.76 = 190 A.
The checks should return to the stated quote and target. A plausible-looking amount is not enough if its currency label or transaction stage is wrong.
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A learner multiplies 200 A by 0.8 when the quote is 0.8 A per B. Ask what the product’s units would represent and what one B costs. The repair is to interpret the rate direction, not simply replace multiplication with division as another unexplained rule.
Another learner reverses 1 A = 1.25 B to 1 B = 1.25 A. The repair is to scale both sides until one B is represented. A ratio table can make the reciprocal relationship concrete.
A third compares gross receipts despite different charges. The repair is to identify the final requested amount and include every stated stage. A fourth compares different starting amounts. The repair is to hold the input or target fixed according to the question.
The marked-paper guide explains how to choose practice from the first line that loses meaning. Exchange questions can involve several distinct errors, so “more money questions” is less precise than “practise reverse quotes without charges” or “practise identifying the fee currency.”
Once a learner can explain the direction, vary the numbers and notation. The aim is to transfer the relationship, not to associate one familiar currency name with a fixed button on the calculator.
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A fictional task asks for a net receipt of 300 B. The conversion rate is 1.25 B per A, and a charge of 10 B is deducted after conversion. Find the A that must be converted to leave the required net receipt.
The gross B receipt must be 300 + 10 = 310 B. Dividing by the conversion rate gives 310/1.25 = 248 A. Forward checking gives 248 × 1.25 = 310 B, then 310 − 10 = 300 B. The check follows the actual order of the stated process.
Dividing the target 300 B by 1.25 first would give 240 A, but that converts to only 300 B before the charge. After the charge, the net receipt would be 290 B. The rate calculation is correct for the gross target but incomplete for the net target.
Now change the fee model. Suppose 4 A are deducted from the total starting budget before conversion, with no charge in B. To obtain 300 B, the amount actually converted must be 300/1.25 = 240 A. The total budget must therefore be 240 + 4 = 244 A.
The two budgets, 248 A and 244 A, answer two different fee structures. Do not compare the fee numbers 10 and 4 without their currencies and stages. In the first model, the charge is expressed in B after conversion. In the second, it is expressed in A before conversion.
A useful backward record starts at the requested net receipt and undoes each stage in reverse order. Undo a subtraction by adding the charge, then undo the proportional conversion using the correctly directed rate. If the fee is before conversion, undo the conversion first and then restore that fee.
This is a mathematical inverse process, not a rule that all exchange services operate this way. The task defines the process. A learner should be able to explain which stage is being undone on each line and which currency the intermediate amount uses.
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An exercise may quote 100 A = 124 B rather than one A at a time. Another quote may state 1 B = 0.8 A. To compare receipts from A, first normalise their directions and their quoted batch sizes.
The first quote gives 124/100 = 1.24 B per A. The second gives 1/0.8 = 1.25 B per A. For a common input of 250 A and no charges, the receipts are 310 B and 312.5 B respectively.
Comparing 124 with 0.8 would be meaningless as a rate ranking because both the direction and the quoted input size differ. Even comparing 124 with 125 would need the statement that both refer to 100 A. The denominator is part of the comparison.
You can avoid forming a unit rate by scaling the quoted pair. Since 250 A is 2.5 times 100 A, the first receipt is 2.5 × 124 = 310 B. This is the same proportional relationship written in a batch form.
Be careful with a table heading such as B per 100 A. A row entry of 124 means the full quoted pair, not 124 B for every single A. Copy the heading into your working before selecting a multiplier. Table labels can carry information that the row’s number does not repeat.
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SECTION 21 OF 22
20. Distinguish a quote comparison from a change-over-time comparison
Suppose an exercise gives an earlier quote of 1.2 B per A and a later quote of 1.25 B per A. For a fixed 200 A, the B receipts would be 240 and 250. The receipt difference is 10 B under the no-charge model.
As a percentage of the earlier B receipt, the increase is 10/240 × 100%, approximately 4.17%. The rate change has the same relative increase because both receipts use the same starting A amount. The numerical difference of 0.05 in the rate is not itself a five-percent increase.
If the question asks for the cost of a fixed B target instead, the A costs fall when B per A rises. For 300 B, the earlier cost is 300/1.2 = 250 A and the later cost is 300/1.25 = 240 A. That is a decrease of 10 A, or 4% of the earlier A cost.
The percentages 4.17% and 4% differ because they describe different ratios with different bases. This is not a contradiction. One compares output receipts from a fixed input; the other compares input costs for a fixed target.
Keep the comparison sentence narrow. Name the quantity, the reference amount and the assumed charges. That makes a change-over-time question another application of the same direction habit rather than an invitation to rank bare decimals or infer a real-world financial outcome from an invented classroom quote.
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Is multiplying always used when converting into another currency?
No. Multiply when the rate is expressed as units of the target currency per unit of the starting currency. Divide when the supplied quote gives the starting-currency cost of one target unit. The labels explain the operation.
Does a bigger final number mean I gained value?
Not merely because it is bigger. Different currency units can represent equivalent amounts with different numerical counts. In the classroom model, interpret the quoted relationship rather than comparing unlike units as bare numbers.
Should I always invert the quote before calculating?
No. You can use the quote as written if you interpret it correctly. Inverting can make comparisons easier when several quotes use different directions, but a common-input or common-target calculation can also work.
Can I ignore a small fee?
Use every stated condition relevant to the question unless an instruction permits an approximation. A fixed fee can change which offer gives the larger net receipt, especially when the exchanged amount is small.
Where should I go next?
Use the Mathematics explanation directory for another focused problem. The Clementi Secondary 1 Mathematics small-group guide gives the local teaching context. The immediate habit is to name the starting currency, name the target currency, and compare complete outcomes on the same basis before ranking any numerical quote.