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Imagine we're going down a hill and we're driving a manual car. There are two situations:
If I put my car in neutral it will go downhill without any need for the injector to squirt fuel into the system, just enough fuel for the engine to idle.
I put the car in the last gear (5th on my car) and let the gravity pull my car.
Which situation will consume less fuel?
Both the title and the last sentence, which I have highlighted, clearly state that I am asking about fuel consumption.
If the car is equipped with a modern fuel injection system, it will likely use less if left in gear and allowed to run against the transmission with no throttle as modern fuel injection systems can and do shut down the injectors completely thus use no fuel whatsoever. If the engine is at idle in neutral, the ECU will have to use a small amount of fuel to keep the engine turning.
If you're looking at just the fuel consumption whilst going downhill, then yes, as Steve Matthews said you'll use less fuel if you're in gear and foot off the throttle - likely no fuel at all.
But you also need to consider after the hill - if you don't brake and allow the car to freely accelerate down the hill, you'll then have considerably more speed than you would if you coasted in-gear. This enables you to use less fuel after the hill levels out. (Safety and speed limits may render this a poor choice!)
It's ultimately a balance between energy loss from being in-gear vs fuel consumption of car "idling" in neutral at speed and energy loss from any braking. If you're not braking with idle coasting and the in-gear energy loss is anything but minimal, then you're likely to be better off neutral coasting. The moment you are braking though, it's likely best to put it in-gear.
Some "hyper-milers" take this so far as to use "pulse and go" - on a level straight, accelerate in-gear then coast in neutral for a much longer period to minimise the in-gear energy loss. Accelerating and decelerating on a straight sounds like it'd be poor for economy, but apparently it works out in certain scenarios.
As Steve Matthews noted, modern cars will not use fuel while coasting downhill in gear. This is called Deceleration Fuel Cut-Off and uses various sensors to determine the engine load (i.e. the car is driving the engine) and whether the throttle is at idle (i.e. the driver's foot is off the pedal). The engine computer will then stop injecting fuel. Generally petrol engines continue to spark, but this consumes minimal energy.
This does not mean the car continues at a steady speed; it will commonly slow down because of mechanical losses in the drive train, and also because work is still being done compressing air on the compression stroke - without combustion, the air acts as a sort of spring, recovering much of this energy, but some heat is lost in the process. Vacuum in the intake manifold (around the throttle plate) of a petrol also causes strong engine braking effect. Diesels do not have throttle plates and their engine braking effect comes mostly from their much higher compression ratios.
Coasting along in gear will allow gravity to drive the car's ancillaries (oil pump, alternator). It also provides drastically greater control than neutral due to engine braking. You can also save fuel by using high-drain equipment such as heaters or air conditioning while descending the hill, as gravity will power these components. This does, of course, detract from the energy available to climb the next hill.
It should also be noted that this applies exclusively to fuel-injected cars; carburetted cars will continue to consume fuel. Additionally, 2-stroke vehicles cannot coast in gear without fuel, as this would starve the crankcase of lubricating oil. However, DFCO has been a feature since the early days of fuel injection - I own a fuel-injected car from the 1980s with this feature.
Technically speaking, coasting in neutral may save you some gas, because mechanical losses in the engine at ~3000RPM are greater than the power needed it run it idle at ~500RPM. Coasting in gear won't consume any gas, but it will slow you down faster, requiring you to reaccelerate earlier than you'd have to when coasting in neutral.
However, if you're in the US, you may have little choice:
Coasting with a vehicle not in gear is prohibited by law in most US states. An example is Maine Revised Statues Title 29-A, Chapter 19, §2064 "An operator, when traveling on a downgrade, may not coast with the gears of the vehicle in neutral.
You'll have to check the laws in your jurisdiction to be sure, but I wouldn't do it anyway. For the fuel economy to become noticeable, you'll have to gain speed while going downhill, which is not a great idea for safety reasons.
TL;DR: On my experience - Long hills on gear, short slopes in the city - on the neutral (not breaking the limits of speed and sanity).
On my petrol powered '97 MMC Galant i've got a custom made trip computer for counting fuel consumption, that is hooked up directly to the fuel injector wires. Here's my observations: when coasting downhill at 4-th gear (i've got 5-speed manual transmission) at around 3000RPM (consuming about 3-4L/100km) - the engine stops sending signal to the injectors only after a 4 to 6 seconds of coasting, which in my city is pretty near to the end of any available slope. So i'd say if you really have an ability to drive in such condition for a certain time (mountain roads, or something) - then it will be a good choice (it will also save your brake disks & pads from overheating and wearing down), but in case of not-so-slopy terrain i'd say that neutral gear will be a greater choice. My car at ~800RPM and 70km\h takes only 0.9-1.2L\100km at neutral gear.
BTW: All said above may be different for any other car, i think modern engine ECUs can activate fuel shutdown much quicker.
This is a really easy question. A vehicle coasting down a hill with with a manual transmission in neutral with the clutch press and held to the floor will burn less fuel than coasting in any gear.
WHY: because an automatic transmission in neutral has a lag on the engine. The Manual transmission in neutral with the clutch press does not have a lag on the transmission cause it it detached from the engine.
