Find your next learning step
Choose the question closest to what you need today, or read the guide from the beginning.
Understand the idea · Compare the examples · See a full solution · Practise and check · Choose the next step
G1 Mathematics is a secondary school Mathematics subject level under Singapore’s Full Subject-Based Banding. This guide explains a practical mathematical decision: comparing offers when the packs, prices and purchase conditions are different. The quickest useful answer is that the lowest price per item and the lowest amount you need to spend are not always the same thing.
Begin by asking what must be bought. Then put the offers on a common basis, check how many whole packs are needed, and apply only the discounts or extra charges that actually qualify. A pack may be better value per item but require a larger payment. A promotion may offer more goods without reducing the cost of the quantity you need.
This is Mathematics about quantities and conditions, not advice to buy a particular product. All prices below are invented teaching examples. The aim is to help students explain a comparison honestly: which option meets the requirement, what it costs and why another attractive-looking number does not settle the decision by itself.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–18
Value asks what each item or unit costs. Outlay asks how much money must actually be paid for a feasible purchase. Suitability asks whether that purchase meets the stated requirement. These questions are connected, but none automatically answers the others. A useful comparison names the question before calculating.
Suppose six items cost $4.80 and eight cost $6. The eight-item pack has the lower unit price: seventy-five cents rather than eighty cents. But if exactly six items are needed and surplus has no use, the six-item pack requires less money. If seven are needed and whole packs must be bought, the eight-item pack becomes the cheaper of those two single-type purchasing routes.
For the wider subject, see G1, G2 and G3 Mathematics explained. Our earlier guide on reading quantities before calculating covers the foundational habit. This article concentrates on fair offer comparisons. Select examples appropriate to the student’s taught work and the 2027 SEC G1 Mathematics syllabus K110.
An offer cannot be judged usefully until the requirement is known. Buying enough drinks for eight people is a different task from finding the cheapest price per litre. A question may specify a budget, a minimum quantity, whole packs, identical servings or a limit on unused goods. Those conditions determine which purchases are feasible.
Read the last sentence of the question before beginning arithmetic. Does it ask which offer has the lowest unit price, which purchase costs least, or which option is suitable under several conditions? Students often answer the first of these because division is familiar, even when the task asks for the second or third.
Write a short requirement statement. “Need at least thirteen items; buy whole packs; mixing pack types is allowed” gives a clear starting point. The statement need not be long. It makes the later comparison answerable and helps prevent an assumption about what a sensible shopper might do from replacing the supplied task.
Do not invent practical restrictions. If a classroom question does not mention travel costs or storage limits, do not silently add them to change the result. In a real decision those factors might matter, but a mathematical exercise needs to be solved under its stated conditions. Identify any necessary assumption clearly.
This first reading step also protects against overclaiming. An option can be cheapest under the given model without being universally best. A final explanation such as “Option A has the lowest total payment for the required six items” is more accurate than announcing that A is always the best value.
Contents · Previous section · Next section
When packs contain different numbers of identical items, divide pack price by item count. Six items for $4.80 gives $0.80 per item. Eight for $6 gives $0.75 per item. The comparison is fair because both results describe the cost of one equivalent item under the stated pack pricing.
The units explain the division direction. Dollars divided by items produces dollars per item. Items divided by dollars produces items per dollar, which can also be meaningful but answers the reciprocal question. A student who obtains a plausible decimal should still identify which rate has been calculated before comparing it with another rate.
Use one basis consistently. Comparing $0.80 per item with twelve items per dollar would mix two directions. Convert both to dollars per item, or both to items per dollar. For the first basis, lower is cheaper. For the second, more equivalent goods for each dollar represents the better numerical rate.
A calculator can help with awkward pack sizes, but it cannot establish whether the items are equivalent. In a classroom problem, assume equivalence only as stated. A large and a small bottle are not identical units of drink volume. If capacities differ, price per bottle may be the wrong comparison basis.
Write the conclusion with its scope. “The eight-item pack has the lower price per item” is justified by the calculation. It does not yet prove that buying that pack is the cheapest way to meet a particular requirement. Keep the unit-rate result available, then continue to the quantity and pack conditions.
Contents · Previous section · Next section
Suppose one cereal pack contains 375 grams and costs $3, while another contains 500 grams and costs $4.20. To compare price per hundred grams, calculate $3 ÷ 375 × 100 = $0.80 for the first and $4.20 ÷ 500 × 100 = $0.84 for the second. The first has the lower unit price.
The choice of one hundred grams is convenient, not compulsory. Price per kilogram would give eight dollars and eight dollars forty. The ranking is unchanged because both rates have been converted consistently. Students should understand why a common basis works instead of believing that only one particular comparison unit is allowed.
Now compare a label giving $1.20 per hundred grams with another giving eleven dollars per kilogram. Convert eleven dollars per kilogram to $1.10 per hundred grams. The second is cheaper per unit mass. Comparing the printed numbers 1.20 and eleven without their units would produce a meaningless judgement.
For liquids, litres and millilitres need the same care. A 750-millilitre bottle is 0.75 litres. If it costs $3, its price per litre is four dollars. A one-litre bottle at $3.80 has the lower price per litre even though its total price is higher than the smaller bottle’s price.
Do the reading and conversion in visible stages during learning. First record each quantity and unit. Then convert to a shared unit. Finally calculate the rate. If an answer is wrong, this layout makes it easier to distinguish a conversion error from a division error or a misunderstanding of the comparison itself.
Contents · Previous section · Next section
A unit rate can describe the pricing relationship, while the shop may sell only whole packs. If a pack contains eight items for six dollars, the mathematical cost of six items at the same rate would be $4.50. That does not mean six items can actually be purchased for that amount under the stated sales condition.
Suppose seven items are needed. One six-item pack is insufficient. Buying two such packs costs $9.60 and provides twelve items. One eight-item pack costs six dollars and provides eight. The eight-item route is cheaper here, but the reason includes feasibility and total payment, not only its lower unit price.
Now suppose six items are needed. One six-item pack costs $4.80, while one eight-item pack costs six dollars. The six-item route has the lower outlay despite its higher unit price. Changing only the requirement changes the answer. This is a useful contrast because it exposes an overly automatic “lowest unit price wins” rule.
Check feasibility before comparing totals. The cheapest listed price may buy too little. A student who selects one six-item pack for seven people has not met the requirement, even if the arithmetic is correct. Ask whether everyone or everything specified can be supplied by the chosen purchase.
The quotient in a pack calculation is an intermediate result. Seven divided by six is more than one, so a single six-item pack cannot suffice. Two are needed if only that pack type is allowed. Interpret the result in relation to whole packs, rather than applying ordinary rounding as an unexplained routine.
Contents · Previous section · Next section
An offer may provide extra items at a lower unit price. Whether that helps depends on the task. If unused goods can be stored and the question asks about long-run unit value, surplus may be relevant. If the task asks for the lowest payment today, unused items do not reduce the amount paid.
Avoid assigning a monetary benefit to surplus without a stated basis. Buying eight items for six dollars instead of six for $4.80 does not save money today. It spends $1.20 more and receives two additional items. That might be desirable under some conditions, but it is not the same as a lower outlay.
Use precise sentences. “The larger pack gives two extra items for an extra $1.20” describes the comparison. “The larger pack saves money because each item is cheaper” leaves out the larger payment. Students should learn to distinguish a valid rate comparison from a claim about the actual purchase.
A hypothetical question may explicitly say that surplus cannot be used. Apply that condition without making a broader claim about real products. The Mathematics then asks which feasible purchase minimises the relevant cost or waste. The conditions define the decision, and a different set of conditions may support a different conclusion.
This is an important form of practical honesty. Mathematical comparisons should not disguise what is being counted as a benefit. Name money paid, goods received and goods required separately. Once those quantities are visible, the learner can explain a recommendation under the exercise’s assumptions without treating every promotional label as a complete argument.
Contents · Previous section · Next section
An item priced at thirty dollars with a twenty per cent discount costs twenty-four dollars. An item priced at twenty-five dollars with a ten per cent discount costs $22.50. The larger percentage reduction does not produce the lower final price in this example because the discounts apply to different original amounts.
Calculate the payable amount for each offer before comparing. For the first, thirty multiplied by 0.80 gives twenty-four. For the second, twenty-five multiplied by 0.90 gives $22.50. The multipliers describe the remaining percentages. They should be linked to the correct original prices, not mixed between offers.
Ask whether the question compares the saving or the final cost. The first offer saves six dollars and the second saves $2.50. The first therefore has the larger reduction, while the second has the lower final price. Both statements can be true. The requested comparison determines which one answers the task.
Do not infer the discount from the final price alone without identifying the original whole. A reduction of five dollars is twenty-five per cent of twenty dollars but only ten per cent of fifty. A percentage is a relationship to a reference amount. The same numerical saving can represent different rates.
During correction, ask the student to label original price, discount amount and payable price. Those labels often reveal where two quantities have been confused. More percentage calculations help only if the learner knows which quantity is being found. The comparison should end with the final quantity requested, not whichever intermediate result is most striking.
Contents · Previous section · Next section
Consider a hypothetical offer: identical items normally cost two dollars each, and buying two at that price provides a third free. Three items then cost four dollars, giving an average price of four thirds of a dollar per item. That rate describes the complete qualifying purchase, not every possible quantity purchased.
If only two items are needed, the qualifying purchase still costs four dollars. A different shop offering individual items at $1.50 would supply two for three dollars. The promotion gives an extra item, but the individual-price option has the lower payment for the required pair under these stated conditions.
If three items are needed, the promotion costs four dollars while three at $1.50 cost $4.50. Now the promotion is cheaper. The requirement has changed, while both pricing arrangements remain the same. Comparing these cases helps students see why an attractive average rate cannot be applied without checking the qualifying bundle.
Do not treat “free” as meaning the whole purchase costs nothing. It means the specified additional item has no separate charge when the conditions are met. The total payment still includes the qualifying goods. Read the full sentence rather than allowing one promotional word to replace the mathematical model.
If a question includes limits, such as one free item per customer, use those limits exactly. Do not assume a three-for-two arrangement repeats indefinitely unless stated. A purchase rule may be conditional or capped. The learner should identify where the rule applies before scaling it to a much larger order.
Contents · Previous section · Next section
Suppose two hypothetical offers supply the same required quantity. Offer A costs ten dollars with no additional charge. Offer B costs nine dollars plus a two-dollar delivery charge per order. For a single order, B costs eleven dollars in total, so A has the lower outlay under these conditions.
The visible item price is not always the full payment. Add every charge the question specifies, and do not invent charges it does not mention. A clear layout separates goods cost, discount and additional charges before producing the final total. This makes the comparison easy to inspect and reduces omitted information.
A fixed order charge is paid once per order, not once per item unless the task says so. If B supplies five items, a two-dollar order charge is still two dollars. Multiplying it by five would change the pricing model. Ask what event triggers the charge and how often that event occurs.
Threshold conditions need the same care. If a fictional offer removes a charge only when eligible goods reach a stated amount, check whether the proposed purchase qualifies. Read whether the threshold is inclusive and what counts toward it. These are conditions in the exercise, not universal rules about real sellers.
The final explanation should show the comparable totals. “A costs ten dollars; B costs nine plus two, or eleven; A is one dollar cheaper for this order” is complete and clear. Simply pointing to the lower listed item price would ignore part of the payment that the question explicitly requires.
Contents · Previous section · Next section
Return to six items for $4.80 and eight items for six dollars. Suppose thirteen items are required and mixing packs is allowed. Three six-item packs provide eighteen items for $14.40. Two eight-item packs provide sixteen for twelve dollars. One of each provides fourteen for $10.80.
The mixed purchase is cheaper than either pure route in this example. A learner who compares only the two unit prices or only purchases of one pack type may miss it. The condition allowing mixtures matters. If mixing is prohibited, the feasible options and final answer change.
Check that the compared candidates cover the relevant small possibilities. No eight-item pack requires at least three six-item packs. One eight-item pack needs at least one six-item pack to reach thirteen. Two eight-item packs already suffice. More packs of either type add positive cost and are unnecessary for finding the minimum here.
This is a small organised comparison, not a demand to test endless combinations. A table helps keep quantity and cost together. Ask the student to explain why an option is excluded: insufficient items, prohibited mixture or higher cost than a feasible alternative. The explanation makes the search understandable rather than accidental.
For larger or more complex choices, use methods appropriate to the course and instructions. The present example is deliberately small enough for direct reasoning. Its lesson is that a rate comparison and a feasible purchase comparison are different tasks, and the purchase conditions determine which combinations need to be considered.
Contents · Previous section · Next section
| Purchase for at least thirteen items | Items supplied | Total payment | Meets the requirement? |
|---|---|---|---|
| Two six-item packs | 12 | $9.60 | No |
| Three six-item packs | 18 | $14.40 | Yes |
| One eight-item pack | 8 | $6.00 | No |
| Two eight-item packs | 16 | $12.00 | Yes |
| One six-item and one eight-item pack | 14 | $10.80 | Yes |
The table prevents a low price from being considered without its quantity. The $9.60 option is cheaper than $10.80, but it does not supply enough items. It is not an eligible answer to the stated requirement. Compare costs only among feasible purchases after checking the quantity condition.
The table also keeps surplus visible. The mixed option provides one unused item; the two-eight-pack option provides three. In this example, the mixed option has both lower cost and lower surplus than the other feasible listed alternatives. Those are separate observations, even though they point in the same direction here.
Ask for a final sentence rather than another row of arithmetic. “Buy one pack of each type, giving fourteen items for $10.80” names the purchase and explains the result. If a question requests justification, include the relevant comparison. The reader should not have to infer which row the student selected.
Contents · Previous section · Next section
A class needs twenty juice servings of 200 millilitres each. Offer A supplies one-litre bottles at $2.40 each. Offer B supplies 1.5-litre bottles at $3.30 each. Bottles must be bought whole, mixtures are not allowed, and no extra charges apply. Find the lower-cost purchase meeting the stated need.
The required volume is twenty multiplied by 200, or 4,000 millilitres, which is four litres. Offer A needs four bottles and costs $9.60. Offer B needs three bottles because two supply only three litres. Three cost $9.90 and supply 4.5 litres. A therefore has the lower total payment for this requirement.
The unit prices tell another true part of the story. A costs $2.40 per litre, while B costs $3.30 divided by 1.5, or $2.20 per litre. B has the lower unit rate. It still costs more for the whole-bottle purchase required here because an additional half litre is bought beyond the four-litre need.
Now change the requirement to twenty-two servings. The required volume becomes 4.4 litres. A needs five bottles, costing twelve dollars. B still needs three bottles, costing $9.90. B is now cheaper. The change in requirement has crossed a pack boundary, which alters the actual payment comparison.
This pair is useful because it prevents a memorised conclusion. Neither offer is always the cheaper purchase for every requirement. The learner must calculate the needed volume, find feasible whole-bottle quantities and compare totals each time. The final explanation should distinguish the invariant unit prices from the changing purchase requirement.
Contents · Previous section · Next section
When a solution fails, locate the first decision that no longer follows from the question. Did the learner compare different units? Choose insufficient quantity? Apply a discount without qualifying? Ignore a fixed charge? Multiply a unit price by a quantity that cannot actually be purchased? These are different errors and need different repairs.
Ask the student to describe what each calculated number represents. A correct unit price may have been mistaken for the total cost. A correct saving may have been mistaken for the final price. The arithmetic can be accurate while the conclusion answers a different question. Labels make that distinction easier to see.
Use a nearby contrast after explanation. If whole packs were ignored, change the need so that it is exactly divisible by one pack size. Then change it slightly above that boundary. Ask why the required number of packs changes suddenly even though the unit price stays the same. The contrast tests interpretation rather than memory.
For unit errors, keep the requirement simple and vary only grams versus kilograms or millilitres versus litres. For discount errors, keep the quantities identical and vary the starting price. Isolate the failed relationship before combining several conditions again. More difficult questions are useful after the basic decision becomes clear.
A useful correction note names the repaired action. “Compare total payments for enough whole packs” gives a future starting point. “Choose the cheapest” restates the goal without explaining how. Students need a method for identifying comparable quantities, not merely a reminder that the answer should save money under the stated exercise.
Contents · Previous section · Next section
These original tasks are informal learning checks. They do not assess suitability for a Mathematics subject level. Ask the learner to state the comparison being made before calculating. Where a condition is specified, use it exactly and explain its effect on the final answer.
One: four identical items cost three dollars, while six cost $4.20. Their unit prices are seventy-five cents and seventy cents, so the six-item pack has the lower unit price. If only four are needed and whole packs are required, however, the three-dollar purchase has the lower immediate outlay.
Two: 250 grams cost two dollars and 400 grams cost $3.40. The prices per hundred grams are eighty cents and eighty-five cents. The smaller pack has the lower unit price. Three: an item reduced from forty dollars by fifteen per cent costs thirty-four dollars. Compare that final price with any competing final price, not merely its discount label.
Four: a qualifying three-for-two offer uses an ordinary price of three dollars each. Three items cost six dollars under the stated offer. If two items are needed, a separate offer at $2.50 each supplies the pair for five dollars and has the lower outlay. The extra item in the promotion does not reduce the six-dollar payment.
Five: eleven items are needed. Packs of five cost four dollars and packs of six cost $5.10; mixing is allowed. One of each supplies eleven for $9.10. Three five-packs cost twelve dollars and two six-packs cost $10.20. The mixed option is cheapest among the feasible minimum-pack combinations for this small example.
Contents · Previous section · Next section
A parent can ask, “What does the question mean by cheapest here?” The learner might identify unit price or total payment. If they are unsure, read the final sentence together. This question guides attention to the goal without announcing the calculation or choosing a purchase for the child.
Another useful question is, “Can you actually buy that quantity under these conditions?” It catches a unit-rate answer that assumes split packs when only whole packs are allowed. Ask the learner to check the number of people or servings supplied before comparing prices. Feasibility should be visible, not assumed.
Keep examples hypothetical or use real labels only as optional practice. Do not turn every family purchase into a test. A brief, calm comparison can make Mathematics meaningful, while constant questioning may make ordinary activities feel like an assessment. Choose moments when the child is willing to explain rather than guess an adult’s preference.
Notice process gains. The student may now convert both capacities to litres, distinguish savings from payment or include a fixed charge without prompting. Praise the specific action and use another example later. This gives a more honest view of learning than declaring the child suddenly good or bad at word problems.
If errors continue, show the teacher a few original attempts with the relevant offer conditions. A statement such as “finds unit price correctly but uses it as the final purchase cost” gives a clear starting point. The teacher can then address the interpretation instead of assigning broad arithmetic practice that may not target the gap.
Contents · Previous section · Next section
A tutor can give the same offers with different quantity requirements and ask each student to make an independent choice. Comparing the results reveals why an offer can be cheaper for one need but not another. The discussion should connect the change to whole-pack boundaries and conditions, rather than treating different answers as inconsistency.
In a three-student lesson, learners can compare unit-rate, table and equation approaches after making their own attempts. Each method needs clear quantities. The tutor can inspect where an individual switches from a true calculation to an unsupported conclusion. That is often the most useful teaching moment in a practical comparison question.
Group size alone does not ensure this quality. Teaching must match the current course, the school’s sequence and the learner’s prerequisites. A student uncertain about decimal division may need that repair before comparing several promotions. Another may calculate fluently but need more work selecting the correct decision criterion.
Our Clementi Secondary 1 Mathematics tutorial guide explains eduKateSG’s small-group approach. Families should confirm current suitability rather than assuming every child needs extra lessons. The useful question is what the support will help the learner understand and do more independently.
A suitable final check changes one condition and removes the previous solution. The learner should explain whether the old conclusion still holds and why. This is better evidence of transfer than repeating the same offer table until its cheapest row is remembered. Good practice develops decision-making from quantities and conditions.
Check whether the units being compared are genuinely equivalent
An item count is a fair basis only when the counted items are equivalent for the question. Five small bottles and five large bottles are the same number of bottles, but not the same quantity of drink. If the task concerns supplying a total volume, compare litres as well as the whole-bottle purchase condition.
Suppose a pack has six bottles of 250 millilitres for three dollars. Another has four bottles of 500 millilitres for four dollars. The first contains 1.5 litres and the second two litres. Both cost two dollars per litre. The lower price per bottle in the first pack does not establish a lower price per unit volume.
If the requirement is one bottle for each of six people, the count may become relevant again. If sharing from large bottles is allowed and the task specifies serving volume instead, total litres may be the better basis. The conditions explain which quantity matters. Neither bottle count nor volume is a universal substitute for reading the actual requirement.
This also shows why a useful comparison can need more than one calculation. A rate establishes value on a common basis, while a feasibility check establishes whether the purchase supplies what the task requires. The final payment then prices the actual whole packs. Keeping these stages separate makes the reasoning easier to inspect.
Ask the learner to name the common unit before dividing. If the answer is merely “each,” ask each what: bottle, litre, serving or pack? That extra word often prevents a misleading comparison. A precise unit is a small piece of language with a large mathematical job, because it identifies what the numerical rate actually describes.
Contents · Previous section · Next section
Is the larger pack always better value?
No. Calculate the relevant unit price rather than inferring value from size. Even when a larger pack has a lower unit price, it may require a higher payment than the quantity needed. Read whether the task asks for a rate comparison or the lowest feasible purchase cost.
Does the largest percentage discount give the cheapest item?
Not necessarily. Discounts can apply to different starting prices. Calculate the amount payable for each qualifying offer. A larger saving and a lower final cost are different comparisons. The example of thirty dollars reduced by twenty per cent and twenty-five reduced by ten per cent shows why the distinction matters.
Should a student always choose the option with least waste?
Only if that is the task or part of the stated conditions. Least surplus, lowest unit price and lowest outlay can point to different choices. A complete explanation identifies the criterion being used. Do not replace the exercise’s requirement with an unstated preference, even if that preference might matter in real life.
Can we mix packs to reach the requirement more cheaply?
If the question allows it, include suitable mixtures in the comparison. If it prohibits mixing, do not use them. For a small example, organise feasible minimum-pack combinations in a table. Explain why insufficient or more costly combinations are excluded rather than trying random quantities until one appears attractive.
What is the most useful first correction?
Ask what each calculated number means and whether the chosen purchase meets the requirement. This often reveals a unit-price result being used as total cost or an insufficient quantity being selected. Repair that exact decision, then use a changed example to check that the learner can make it independently.
Contents · Previous section · Next section
SECTION 18 OF 18
17. The Next Step: Compare the Same Thing Under the Same Conditions
A fair offer comparison keeps the requirement, unit basis and purchase conditions visible. The student can then distinguish cheaper per item from less money paid, and distinguish an extra quantity from an actual saving. The final choice becomes a reasoned response to the task rather than a reaction to the most striking price label.
Choose one school question and write three short lines before calculating: what is needed, what each offer supplies and what conditions apply. Then compare feasible totals and explain the result. If the unit-rate conclusion differs from the purchase-cost conclusion, identify why. That difference is often the most useful learning in the question.
Continue to How Mathematics Works for related learning routes. All offers, prices and purchase conditions in this guide are original hypothetical examples. Course context was checked against the linked SEAB G1 Mathematics syllabus on 15 October 2026; use the student’s taught scope and applicable examination year for formal requirements.
Contents · Previous section · Mathematics hub
Continue the G1/G2/G3 Mathematics series
What Is G2 Mathematics | Why a Fixed Starting Amount Breaks a Proportion Shortcut
What Is G3 Mathematics | How Removing One Value Changes the Mean and Median
What Is G1/G2/G3 Mathematics | Why Show That Needs Working Even When the Answer Is Printed
Earlier guide: Read scale marks and intervals
Continue through the Mathematics learning hub
Browse all 16 G1, G2 and G3 Mathematics explainers for a guide matched to the next difficulty.